Does a Minor6 chord with the 6th in the bass exist or do you just call it a m7b5?

I noticed that for example a Cmin6/A is enharmonically the same as a Am7b5, I completely understand calling it a Cmin6/A in the context of like descending bassline or line cliche situation, but other than that does a Cmin6 with the 6th in the bass actually exist or should it technically just be called a Am7b5?