How is modulo invertible in the general case? If it is, can you demonstrate? I have picked certain numbers and their values modulo 10 are 6, 7, and 8. What numbers did I pick?
It doesn't have to be. Onewayness says nothing about finding "the original x". If a function is one-way, it means that given y (a random n-bit string in the output space of h), it is hard to find any x such that h(x) = y. If your h() is just a modulo operation, this is trivial: just choose x = y.
Ah, I see, way as in path, not way as in direction (which is I think the misinterpretation that leads people to the root of the misunderstanding). So, there's one path to get from x -> y, but it does not imply you can't get back to the original x. In that respect, "way" is an unfortunate word to use, at least as it's used in modern English.
You can demonstrate that a number input into modulo belongs to a specific group. Even not knowing the specific value of modulus you can drive it by observing statistical properties of the output. (which is why modulo hashing gives many collisions)