Most horses do have 4 natural gaits but there are a few specialists. I am not familiar with the Tennessee Walker but there are the harness racing horses who "pace" by alternating both left legs with both right legs. (I used to ride a retired pacer who was learning to canter - his front legs cantered and his back ones paced. Weird ride!)
There is also the passo (S Amer) or tolt (Icelandic) which is supposed to be so smooth that you can ride with a full wineglass and not spill any wine. Less smooth after the fourth or fifth ride, I'll bet.