Does NEC allow for recessed lighting to be placed in a vertical wall?
As noted by Skaperen and Longneck in comments below, NEC requires using listed devices in accordance with manufacturer instructions. So if it says for ceilings, the code requires that you only use it in a ceiling. All of the can type recessed fixtures I have seen are described by their manufacturers as ceiling fixtures (but there may be versions out there that do not have this limitation). If you proceed, you may have a code violation and you might get pushback from an inspector. 

If you do decide to go ahead with this approach, you need to make sure that there is sufficient room in the wall to accommodate the unit. Most stud bays are either 3.5 or 5.5 inches deep and most fixtures are deeper than that (usually 7+ inches).

You do mention new construction units, which have braces reaching the studs. If you have the wall open, or are willing to do that, the support brackets should easily hold the unit steady (although there may be some twisting pressure on the bracket arms).

There are can fixtures specifically intended for slope ceilings. They tend to be new work versions with mounting brackets, not remodel versions with spring clips. If mounted on a wall, they may give you a downward tilt that fits your goal.

If you consider standard remodel recessed ceiling fixtures, the problem you may face is that the mounting brackets for most use spring clips to hold the unit in place.  In a ceiling mount, the weight is evenly distributed around the rim on all of the clips.  If you mount these types of units in a wall, the weight of the unit will pull down in the back, tending to push the bottom of the fixture out.

The springs may be strong enough to hold the unit flush, but you would probably have to try one on a sample of dry wall to see. The Halo brand unit talks about driving a screw through the shell to lock in the spring, and that might help. Other brands may have similar features.In sum, I don't think there is a code issue, but you may get local reaction and you need to make sure you can properly mount the unit solidly.

There trim kits that are intended to be wall wash baffles which cast light to one side (on your case, downward) that might work well. Also, smaller units, such as 3 or 4 inch versions, will weigh less and may pose less mounting problems. Also be sure that you use insulation contact versions if there is insulation in the wall that may come in contact with the shell. 

You also might consider small, flush mount or recessed LED units that are significantly shallower than conventional recessed lighting. These also may not be restricted to ceiling mounting (or horizontal mounting), and may be more likely to be code compliant.