Memory foam is heavy, dense, and floppy, and it holds shape poorly when handled, so it fights you if you try to carry it upright. Keeping it flat and bagged is the trick, and re-compressing a bed-in-a-box is best avoided.
Pro tip: If you must fold it, fold with the sleep surface inward and strap it, then unfold it flat the moment it reaches the new room.
Watch out: Do not leave a memory foam mattress folded or compressed for days; prolonged folding can crease the foam and it may not fully recover.