That is one of my favorite (most touching) parts of very formal ceremonies too... As you can see below, the tack and fittings are very specific and according to tradition. But in the particular case of Reagan, if I were in charge, I would have the horse carry Reagan's own saddle and boots. (I did hear that the horse will be carrying Reagan's boots)
Military District of Washington - Fact Sheet: The Caparisoned Horse
The Caparisoned Horse
The practice of having the charge of a deceased military officer led in the funeral procession is a survival of an ancient custom of sacrificing a horse at the burial of a warrior. Generally the horse was hooded, sheathed in a cloth or armored covering and bore a saddle with the stirrups inverted and a sword through them.
This further symbolized the fact that the deceased had fallen as a warrior and would ride no more.
During the period of Genghis Khan and Tamerlane, the Mongols and Tartars believed that the spirit of a sacrificed horse went through "the gate of the sky" to serve its master in the after-world. According to European folk belief, the dead horse would find its dead master if permitted to follow him into the hereafter. Otherwise, the dead master's spirit would have to walk. When Gen. Kasimer was buried at Treves as late as 1781, his horse was killed and placed in the grave with the dead general.
Some of the Plains Indians in America adopted the custom after they came into possession of horses.
Horses are no longer sacrificed in such cases, but sometimes a riderless horse is still led in the funeral procession as a symbol of a fallen warrior. In about 1800, Blackbird, an Omaha chief, was buried sitting on his favorite horse.
According to historical records, Abraham Lincoln was the first President of the United States honored by the inclusion of the caparisoned horse in his funeral cortege. When his body was taken from the White House to lie in state in the Capitol rotunda, the casket was followed by the dead president's horse with its master's boots backwards in the stirrups.
In order for the caparisoned horse to be used, the person it is honoring must have at one time been an Army or Marine Corps colonel or above.
Since the president of the United States is the commander-in-chief of the armed forces, he is entitled to the use of the horse.
EQUIPMENT FOR THE CAPARISONED HORSE
The equipment a caparisoned horse bears differs according to its color: if black, the horse carries saddle blanket, saddle and bridle; if any other color, the horse carries a hood and cape, along with a blanket, saddle and bridle.
CAPE: The cape buttons at the breast (four buttons) in front of the left leg. It has a crupper that fits under the horse's tail to hold the cape secure. A crupper is a padded leather strap that is passed around the base of a horse's tail and attached to the saddle or harness to keep it from moving forward. The cape is bordered with a fringe, 3 inches in length, with a 6-inch tassel, spaced every 4 inches. The cape hangs to the hock (the joint bending backward in the hind leg) and knees.
HOOD: The hood covers the head, going back as far as the withers (the highest part of the back of the horse). It buttons under the jaw bone, along the neck to the breast. The hood has eye slots and extends down the edge of the mouth. It covers the ears, and the ear pieces are fringed. Its bordering is like the cape.
SADDLE BLANKET: The saddle blanket extends from the withers to the flank. In width, it extends half-way down the side of the coastal region. There is a white border 1 1/2 inches in width completely encircling the blanket. Stars are placed on the rear corners of the blanket (four inches from the bottom) for generals, with the number of stars indicating the rank.
SADDLE: A pair of spurred boots is placed backwards in the stirrups of the saddle, the tops of which are fastened to the stirrup strap. The officer's field saber and carrier is placed on the "off side" of the horse. The carrier is fastened to the saddle, and at the bottom there is a strap going under the horse's abdomen fastening on the "near side" to the cantle (the upward-curving rear part of the saddle) by straps and a D-ring. This keeps the saber vertical.
BRIDLE: The bridle consists of a snaffle bit (light and jointed, attached to the bridle and having no curve) and a French halter. It is worn in the regular manner, with one rein. This rein is secured to the pommel (the rounded, upward-projecting front part) of the saddle. The horse is led from the "near side" with the ring hand grasping the reins, six inches from the bit.
MISCELLANEOUS: All of the caparison (the ornamented covering) is black. The hood, saddle blanket and cape are made of wool or serge (a strong, twilled fabric with diagonal rib). All brass and leather is highly polished.