In number theory,
Terry Tao already mentioned Quadratic Reciprocity in his first comment,
but there's also the 
<a href="http://en.wikipedia.org/wiki/Dedekind_sum#Reciprocity_law">reciprocity
formula</a>
$$
s(b,c) + s(c,b) = 
 \frac1{12}\left( \frac{b}{c} + \frac1{bc} + \frac{c}{b} \right) - \frac14
$$
for Dedekind sums, symmetrized further in
<a href="http://en.wikipedia.org/wiki/Dedekind_sum#Rademacher.27s_generalization_of_the_reciprocity_law">Rademacher's
formula</a>
$$
D(a,b;c) + D(b,c;a) + D(c,a;b) = \frac1{12} \frac{a^2+b^2+c^2}{abc} - \frac14.
$$
[Here $D(a,b;c) = \sum_{n\,\bmod\,c} ((an/c)) ((bn/c))$,
where $((\cdot))$ is the sawtooth function taking $x$ to $0$ if
$x \in {\bf Z}$ and to $x - \lfloor x \rfloor - 1/2$ otherwise;
and the Dedekind sum is the special case $s(b,c) = D(1,b;c)$.]