Rest Day Question

Options
I have a quick question regarding rest days. Can I do cardio on my rest day so long as it doesn't use any muscles I've previously worked?

More details below:

Currently I have a 3 day cycle that I like to go through. I have an upper body pull day (mostly upper back), a core day (with a little bit of cardio), and a push day (mostly chest). Each day I lift for about 45min-60min and I generally do wave sets consisting of two waves of 12 reps, 10 reps and 6 reps. Usually I will go through my 3 days, rest a day, then start again.

My question is, could I replace this rest day with a more rigorous cardio activity, utilizing my lower body that I do not train otherwise?

I am trying to burn some extra calories in addition to keeping a good diet, so I was considering adding an HIIT workout in on that fourth day.

I am currently under the impression that if you let every muscle recover properly before working it out again, then injury can be avoided and gains will be made effectively. However, I have read many people claiming rest days are important in a proper workout structure.

Thanks in advance!