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Is there a general concept for the congruence of a congruence?

Suppose we have three arbitrary integers $N, m$ and $n$. My question is whether $(N \bmod n) \bmod m$ corresponds to any known mathematical concept (beyond a double operation).

A simpler case, for $m=2$, would be: Is $N \bmod n$ even or odd? Is there any result for this kind of operation?

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There are two uses for "mod" and they often get confused. The computer operation "mod" is a unary operation and it's just finding the remainder upon division. This is the sense you're using "mod" and I wouldn't call that a "congruence."

The more math-y usage is as an equivalence relation and "mod" is really part of the $\equiv$ symbol (or a modifier of it.) $a \equiv b (\bmod{n})$ is a "congruence" because you have two things and they're congruent to each other..

So your question is really "can I divide $N$ by $n$ and then divide the remainder by $m$?" The answer is "sure."

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    The question was not worded correctly. In fact, I would prefer to know results about congruences: can we say anything about the properties of two numbers that are congruent modulo n? If they are they congruent to other integers modulo m do these integers have any known relation to the first integers? – 2017-01-17