You can change a mob's y rotation via /teleport, but not the x rotation.

Steps to reproduce:

1. Summon a mob in a new world with no mobs. (e.g. use a zombie)

2. in a repeating always active command block, put in /execute @e[type=zombie] ~ ~ ~ teleport @s ~ ~ ~ facing @p

3. to better visualize the facing, summon a new entity (e.g. armor stand)

4. in a repeating always active command block, put in /execute @e[type=zombie] ~ ~ ~ teleport @e[type=armor_stand] ^ ^ ^1

5. Watch the armor stand move in circles as you move around the mob in a circle, but not move up when you move up.