Consider functions of $x$:
$$ 1,\quad \sin \frac{\pi n}{N} x, \quad \cos \frac{\pi n}{N} x, \quad \cos \pi x $$
where $n$ runs $1,\,2,\, \ldots,\,N - 1$
Those functions are orthogonal if the inner product is defined as $$ \int_0^{2 N} f(x) g(x) \, dx $$
but also if it is $$ \sum_{x = 0}^{2 N - 1} f(x)g(x) $$ What is special about the sine and the cosine? Do other sets of functions have a similar property.
As @badjohn suggested I experimented with four triangular functions:
The integrals and the sums are zero. The sums are zero also with a shift ($0 \le \delta < 1$, $N = 2$):
$$ \sum_{x = 0}^{2 N - 1} f(x + \delta)g(x + \delta) $$