If your mat doesn't have lines on the back, try cutting with that side. If your cutter works fine, it is the lines on the mat that are your problem. I remember someone (either F&P or Jenny Doan) talking about this on one of their shows or tutorials. I have a couple dark blue mats and purple mats that are 12x18 and a larger (18x24) mat that is yellow on one side and lavender on the other and don't have any problem with any of them. I tried a friend's green mat, but for some reason, I had problems "seeing" with the mat. Maybe it was the yellow lines. I know I can't use Olfa rulers because of the yellow markings.
Well, I just re-read your original post and it sounds like one of those June Tailor's pressing mat on one side and cutting mat on the other. I bought one of those years ago with a gray mat and never had a problem. I just won one of the new, blue ones, but haven't used it yet. I'll have to try it out and see if I have a problem.