Compare two integers in C or C++ without comparison operators

90 bytes

If we can use stdio, why not use its formatting capabilities to perform comparison?

main(a,b){scanf("%d%d",&a,&b);snprintf(&a,2,"%d",b-a);a&=63;putchar(51-!(a-45)-!!(a-48));}

Assumes ASCII-compatible encoding and little-endianness.

72 bytes

Quotients are rounded toward zero but right-shifts are (in practice) "rounded down". That's a dead giveaway.

main(a,b){scanf("%d%d",&a,&b);a-=b;putchar(a?a|=1,a/2-(a>>1)?60:62:61);}

65 79 bytes

Another distinguishing property of negative numbers is that they produce negative modulo. This one doesn't depend on integer representation at all; it even works on my 8-bit excess-127 toaster! Oh, and since we can use conio, why not save two bytes with putch? Now, if I could only find my copy of TurboC...

main(a,b){scanf("%d%d",&a,&b);long long d=a;d-=b;putch(d?d|=1,d%2-1?60:62:61);}

EDIT: Handle large differences assuming long long is wider than int.

Maybe I'm missing something in the rules, but...

81 bytes

main(a,b){scanf("%d%d",&a,&b);long long l=a;l-=b;printf("%lld%d",--l>>63,l>>63);}

Ouputs 00 if a > b, -10 if a == b, and -1-1 if a < b.

64 61 characters

main(a,b){scanf("%d%d",&a,&b);for(a-=b;a/2;a/=2);putchar(a);}

Prints the character values of -1, 0, and 1 for less than, equal to, or greater than, respectively.

This implementation relies on undefined behavior for b being of type int and for inputs outside the range INT_MIN / 2 to INT_MAX / 2. On platforms where signed overflow wraps around, whether 2s-complement (basically all of them) or sign-magnitude, it will fail for 25% of possible pairs of valid int. Interestingly (to me anyway), it will work correctly on platforms where signed overflow saturates.