Congruence of a congruence $(N \bmod n) \bmod m$ 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?
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."
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."