Background
What I think of Ehrhart theory (http://en.wikipedia.org/wiki/Ehrhart_polynomial) asserts that if we take a lattice polytope $P$, and count the number of lattice points in the $t$th dilation of $P$, the result is polynomial in $t$.
If, however, $P$ is only a rational polytope, then in general we get a quasi-polynomial; that is, there is some period $N$, so that if for any $k$, I only look at $t$ congruent to $k$ mod $N$, then again I have a polynomial behavior; however, different residue classes will in general result in different polynomials.
One might expect that any two distinct residue classes have distinct polynomials, but this need not be the case: two different residue classes can have the same polynomial. These are the "accidental" equalities of the question title.
A simple example
For a simple example, take the triangle bounded by $x=0,y=3x$ and $y=1$; scaling by $t$ just changes $y=1$ to $y=t$. The count is quasi-polynomial of period 3, but only has two distinct polynomials. The number of lattice points is
$$ 1+\frac{t^2+5t}{6}$$ if $t$ is congruent to 0 or 1 mod 3, and $$ 1+ \frac{t^2+5t-2}{6}$$ if $t$ is congruent to 2 mod 3.
Questions, general and specific
I'm interested in what we know about when or why these accidental equalities occur. That is a rather broad and open-ended question; so here's something a bit more specific.
If instead of just counting the lattice points in the $t$th dilate, we sum a polynomial function over the lattice points, we again get (quasi-)polynomial functions; call this Euler-Maclaurin theory. I had the naive hope that "Accidental" equalities in Ehrhart quasi-polynomials might extend to equalities in Euler-Maclaurin theory, but this appears not be always the case: if we try to sum the function $x$ over the lattice points in the $t$th dilate of the polytope in our example, we get $$\frac{t^3}{54}+\frac{t^2}{9}+\frac{t}{6}$$ for $t$ congruent to 0 mod 3, but $$\frac{t^3}{54}+\frac{t^2}{9}+\frac{t}{18}-\frac{5}{27}$$ for $t$ congruent to 1 mod 3.
If I have some rational polytope that has accidental equalities in its Ehrhart polynomials, and a specific polynomial I want to sum over it; is there some conditions in which the same accidental equalities will hold for the Euler-Maclaurin problem?
