Legally yes, but it would be very stupid, and no club that I've ever flown with would permit it.
I beg to differ.
If someone has, say, 30 hours in PA28, and a like amount in C-152s, I see no particular stupidity in a recurrency checkout in one re-qualifying that person to fly both. After all, they are both simple, easy-fly airplanes ... it's not like Pilot16 is asking whether a C152 checkride will also qualify him to operate a Globe "Super Swift".
I do agree that if someone has never flown a particular type before, it is only common sense to undergo a checkout in that model, regardless of its simplicity.