In the summer of 1833 there was a great deal of activity being generated by the raising of the 1st US Dragoon Regiment, then being formed at Jefferson Barracks, Missouri. Unlike most equipment acquisitions for mounted units in later years, the dragoon regiment equipment, uniforms and weapons came from a wide variety of sources, mostly private contractors. Many of the component pieces such as halters, bridles, etc. were purchased from regular civilian harness and saddlery suppliers from existing civilian styles. Indeed, the dragoon saddle selected and approved by Lt. Col. Stephen W. Kearney was a slightly modified civilian Spanish saddle, acquired from a local saddlery of note operated by Thornton Grimsley.  They were purchased from Thornton Grimsley in St. Louis until 1836, after which they were made primarily in Philadelphia. The initial saddlery orders for the 2nd Dragoon Regiment were made with John Young & Co. 
The description of this saddle can be read in the actual contract letters – later sources that described additional details (usually complaints about them) are included and summarized here. The tree was the classic Spanish type, made with four wooden pieces, covered in rawhide.
In the fashion of the early frontier, the bottoms of the sidebars were padded, with heavy fabric or canvas stuffed with deer hair. These caused no end of annoyance to the troopers far afield and many would end up removing them, though they were retained for most of the service life of the design. Col. Henry Dodge, the first commander of the regiment, mentions this: ‘the pads for the dragoon saddles became too heavy on long expeditions and often had to be thrown away. “The Indians,” he observed, “Seldom Have anything under their saddles but Blanketts or Skins and they seldom Hurt there Horses Backs.” ‘ 
Single girth straps, 1 1/4″ side, were attached to each side the tree with a small leather safe between them and the tree pad. This suggests that the pad extended out from underneath the tree for a small distance. The contract description notes that this safe is to protect the pad from the buckle and girth, which tells us that it probably extended for some distance from the seat.
A description of the girth and surcingles was provided in a letter from March, 1838: “The surcingle and girth are component parts of the saddle and those at present used are of black leather according to sample received by me but they are rather stout and heavy being in the way of the horseman (the surcingle I mean) and the color is exceptionable as the texture of the leather is injured by the coloring matter employed, copperas.” 
Stirrups would hang from a triangular iron wire loop, attached to the tree. Placement of these on the tree isn’t exactly known, though most illustrations of similar spanish saddles and trees show them at the base of the pommel, or slightly to the rear of that point. Later photographs and examples of similar spanish tree saddles show placement of stirrups near the center of the seat, so it may have varied at time went along.
Large skirts were attached to the tree, that would naturally cover all the girth strap, safe, and extend far enough to give the dragoons uniform trousers some protection.
To provide for baggage attachment to the cantle, three iron staples were driven into the rear cantle face, and clinched. One center staple for the crupper, and the other two to the sides for buckling the coat pad to the seat. The coat pad is described as “the usual form with two straps & buckles”, which is somewhat not helpful, as this common items did not last in general use, and was certainly gone by the early 1840s in military usage. The valise would be strapped to the coat pad, which would be attached to the seat. What you have then is a over-stuffed roll with a sweat-soaked deer-hair stuffed pad, bouncing around on the horse’s kidney area – it wasn’t a popular item.
By 1835, regular saddlebags became much more common, to the near exclusion of the valise. Henry Dodge mentioned it in his ‘Military Orders’ journal that the saddlebags in use were entirely satisfactory, this in 1836.  The 2nd Dragoon equipment purchase from John Young & Co.  only refers to saddlebags – no valise or coat pads. There was a later letter found, dated September 18, 1838, concerning the ‘component parts of the saddle’, for contract purposes, ie., what was part of “the set”. No valises, but the coat pad still remains… “The component parts of a saddle complete agreeable to the present contract are one pair of stirrup leathers and irons, coat pad and straps, crupper, girth and surcingle.” 
The saddle was provided with a breaststrap (sometimes referred to as a ‘martengale’), that attached to the saddle by buckling to ‘leather loops’ attached to each side of the pommel. A leather loop can be two things – an actual standing loop, or a short length of leather strap that is nailed on each end, with enough space between it and the tree for the end of the breastplate strap to pass under. This latter type sounds much more plausible, as a billet end could pass under and be brought back over to a buckle on a chape stitched to the breaststrap.
“Chain halters” were issued and tried after 1833 – a full complement being issued to the 2nd Dragoons in 1836. No imagery has been found to suggest what these might have been like. They seem to have fallen out of favor by the end of 1840, as requests for material that year only mention leather halters being desired for consideration in the coming year.
Two leather straps with buckles were attached to the tree very near these ‘loops’, to hold the pipes of the pommel holsters. Pommel holsters were of the usual type, with leather flaps covered in black bear skins. The connecting piece between the two sides, called the seat, would have a large hole in the center to go over the saddle horn. Officers in later correspondence noted the tendency of holsters to bounce and occasionally slide up and off the horn during animated gaits – with the usual remedy being to simply nail them to the pommel.
In this initial order made in 1833, many of the components and parts were made in Philadelphia and shipped out to Jefferson Barracks. 1,000 pairs of spurs and straps were purchased from J. Carrigan, Jr. 1,000 bridles, whips, pairs of stirrup leathers, surcingles, girths and cruppers were obtained from William S. Hansell.  Stirrups were brass, with a three-bar picked tread, and a transom loop top – 1,000 pair were cast in Philadelphia by William R. Smith. 
As far as the hardware used in the stirrups straps, holster pipe straps, coat pad, crupper, bridle, halter, etc. – when these were even mentioned they were ‘iron buckles’. It’s reasonable to presume they were the common iron horseshoe shaped buckles so common at the time.
Wool saddle blankets were issued in brown, or dark blue. A Commissary General letter from April, 1836 provides detail: “I will thank you to have made and forwarded to this office as early as will be perfectly convenient, 15 or 20 blankets suitable for saddle or horse blankets for the U. S. Dragoons. I want them as a sample upon which to found contracts hereafter for the article or purchases as the case may be. The foreign article of proper dimensions is flimsy, light and by no means durable. About 700 of these are annually required for the Regiment of Dragoons. It is not necessary that the wool should be very fine for horse blankets. Each blanket should be 6 feet long by 5 feet wide, and weigh about 3 pounds. State the price at which such blankets can be furnished. The blankets should be of a dark color, a deep brown or a blue will answer if durable. Blue will be preferred.”  Lt. Col. Stephen W. Kearney, in early 1837, requested blue blankets with yellow edge stripes, and company letters on each corner – the commissary general calmly stated that the blanket contract for the year had already been let, so he would be getting blue ones.
Col. Kearney was set on having his company letters, and the next year a comment was elicited in communications by the Commissary General C. Irvine that clearly showed his frustration with this sort of inanity: “Permit me to remark that frequent changes in the military supplies and equipments as the suggestions of officers of the army whose tastes and opinions will be different until now shall be differently constituted are injurious to the service. Comparable improvements should be adapted without hesitation. All military equipment should be neat and simple in their construction. There should be nothing complex about them nor so fashioned as to create delay in their provision or issue. If Colonel Kearney’s suggestion shall be adopted in relation to lettering the horse blankets for each company of dragoons it will be necessary to have at all times on hand in advance a duplicate or triplicate supply of them to guard against delays in the issues. This remark will apply also to the suggestion of lettering the knapsacks.” 
That summarizes the basic description of the 1833 dragoon saddle and related equipment. Now we'll take a look at a collection of images of this equipment in action...
Well, there isn't any. The 1833 dragoon equipment while it was in active use by the dragoon regiments (1833-1844 or thereabouts) pre-dated nearly all photographic processes. No examples are known to have survived.
So, what do we have that can tell us what the form and look of a dragoon 'spanish saddle' might have been? We have some visual images that can inform us what a spanish saddle looked like, and a couple of photographs that can give us a potential peek.
The difficulty for most researching this subject is that there is almost no photographic evidence of this saddle, and it's near-lookalike, the wagon drivers saddle adopted in 1847 and used thru 1875.
However, through the generous sharing of a forum member, a fine image has come to my attention that shows what is very likely one of the only clear photographs of a 'dragoon saddle' in actual use. While this image was taken in 1850, a few years after the 1833 dragoon was replaced by the Grimsley dragoon equipment, it's form and composition is visually identical to the descriptions of the 1833 dragoon equipment. This particular saddle was used on a military surveying expedition by a naturalist/doctor who was soon commissioned. There are numerous references in contemporary accounts that speak of travellers using 'an old dragoon saddle', which was nearly synonymous with 'old spanish saddle'. And just because these facts are important, the horse's name was 'Davy'. We appreciate Davy taking a siesta while the photographer was doing his thing.
Regarding the '1833 Dragoon Saddle' as depicted in Randy Steffen's "US Military Saddles" and "US Horse Soldier" series of books, let us be direct and frank in our discussion of them. They are wrong, not even close - just stop it.
If you want to know why or start a 'discussion' about it, you can see a detailed examination of it here.
 Congressional Series of United States Public Documents, Volume 298, 1837.
 Col. Henry Dodge's 'Military Orders' journal - from paper read before annual meeting of the Mississippi valley historical association in St. Louis, May 8, 1919. by Louis Pelzer.
 Com. Gen. Callender Irvine to Major L. Whiting, U.S. Army Clothing Bureau. National Archives, Record Group 92, Entry 2117, Commissary General of Purchases Letters Sent (Masterson 35), Book O. Research courtesy Stephen E. Osman
 Com. Gen. Callender Irvine to Major H. Whiting, U.S. Army Clothing Bureau, National Archives, Record Group 92, Entry 2117, Commissary General of Purchases Letters Sent (Masterson 35), Book O. Research courtesy Stephen E. Osman
 Com. Gen. Callender Irvine to to Richard Kimball, Rochester, New Hampshire, National Archives, RG 92, Entry 2117, Commissary General of Purchases Letters Sent, Book N. Research courtesy Stephen E. Osman
 United States congressional serial set. 256 (1833/34), pg 36.
 United States congressional serial set. 256 (1833/34), pg 35.
 'The National Gazette', 28 Dec 1840, Philadelphia, Pa.