Is there an expression using modulo to do backwards wrap-around (“reverse overflow”)?

Your expression should be ((x-1) + k) % k. This will properly wrap x=0 around to 11. In general, if you want to step back more than 1, you need to make sure that you add enough so that the first operand of the modulo operation is >= 0.

Here is an implementation in C++:

int wrapAround(int v, int delta, int minval, int maxval)
{
  const int mod = maxval + 1 - minval;
  if (delta >= 0) {return  (v + delta                - minval) % mod + minval;}
  else            {return ((v + delta) - delta * mod - minval) % mod + minval;}
}

This also allows to use months labeled from 0 to 11 or from 1 to 12, setting min_val and max_val accordingly.

Since this answer is so highly appreciated, here is an improved version without branching, which also handles the case where the initial value v is smaller than minval. I keep the other example because it is easier to understand:

int wrapAround(int v, int delta, int minval, int maxval)
{
  const int mod = maxval + 1 - minval;
  v += delta - minval;
  v += (1 - v / mod) * mod;
  return v % mod + minval;
}

The only issue remaining is if minval is larger than maxval. Feel free to add an assertion if you need it.