If your door is 32 bits wide, and you have to carry a mattress that's 64 bits through it.... can you do that holding it normally? No.
What if your door is 64 bits wide, and you have a mattress that is 32 bits wide? Of course you could!
Same goes for basically every situation this problem crops up. 32x software on a 64x OS? Can do! 64x software on a 32x OS? Nope.
32x OS on a 64x Processor? Can do! The opposite... not so much.