Any horse should know a bit about neck reining. Just enough to get by is fine, but to have a truly well rounded education, this is pretty important, even on a horse who will not be used in the Western world.
One way to train the horse to neck rein is to start by using a simple direct rein. This is what you started with when you trained your horse to respond to the reins. Now, instead of just opening the left rein to go left and the right rein to go right, start opening the left rein, but adding pressure with the right rein on the side of the neck to go left. Gradually you can decrease the direct rein and increase the neck rein until your horse uses only the neck rein.
The next part will be a different, but more difficult (and somewhat faster) way to train the horse the same thing. Until next time, safe riding!