For example for mod 11 I get that all the congruence classes of it has an inverse pairing:
1 (mod 11) and itself,
10 (mod 11) and itself,
2 (mod 11) and 6 (mod 11),
3 (mod 11) and 4 (mod 11),
5 (mod 11) and 9 (mod 11),
7 (mod 11) and 8 (mod 11)
As you can see for every congruence class you can find an inverse pair.
Is it true for every number?
If so, why?
No, it is not true for every number. In fact, it's true if and only if the modulo is of a prime number. Anytime you have a non prime modulo, numbers that are not relatively prime to the modulo do not have an inverse, as they are zero-divisors in the ring.
For example, in mod 6, 2 and 3 are zero divisors, and there is no inverse for either of them.