How to reorient the facing direction of a 2.5D character Arcade side-scroller?

If you have ever seen or played old 1980’s arcade games such as Dig-Dug or Mr. Do, you know that the character faces the direction that they are moving. If he is moving right, he faces right, left he faces left. Obviously, this is simple and already in a side scrolling template. However, what if the character is facing up or down? How can they be orientated to face the direction they are moving?

Examples:

And:

I have tried to do this with vector manipulations, but they all ended up in Gimbal lock or some sort of issues with movement. My goal is to recreate such classic arcade games in 2.5D and this step is a huge roadblock. Does anyone have any helpful insight as to how I can achieve this functionality?