> I guess my question is: why do you think I am throwing
> away a result which you only accomplish differently?
> "++" vs. "+ 1" -- they both add one to the value
> of $flip.

Um, no, they do not. Saying

$flip + 1;
[download]

does not change in any way the scalar referred to by $flip, while "++$flip" does. Simple as that.

===
What, fundamentally, is the difference between 
  $flip = ++$flip % 4
        and 
  $flip = ($flip + 1) % 4
===

The first changes the value of $flip twice, the second, only once. In the first one, the computer is saying:

1. Increase the value of $flip by one
2. Modulo 4 it
3. Assign the result into $flip

versus:

1. Compute the value of $flip + 1
2. Modulo for that (temporary) value
3. Assign the result into $flip