  We produce a new proof of the reciprocity law for the twisted second moment
of Dirichlet L-functions that was recently proved by Conrey. Our method is to
analyze certain two-variable sums where the variables satisfy a linear
congruence. We show that these sums satisfy an elegant reciprocity formula. In
the case that the modulus is prime, these sums are closely related to the
twisted second moment, and the reciprocity formula for these sums implies
Conrey's reciprocity formula. We also extend the range of uniformity of
Conrey's formula.
