  Let G be a finite group and p be a prime. We investigate isomorphism
invariants of $\mathbb{Z}_{p}[G]$-lattices whose extension of scalars to
$\mathbb{Q}_p$ is self-dual, called regulator constants. These were originally
introduced by Dokchitser--Dokchitser in the context of elliptic curves.
Regulator constants canonically yield a pairing between the space of Brauer
relations for G and the subspace of the representation ring for which regulator
constants are defined. For all G, we show that this pairing is never
identically zero. For formal reasons, this pairing will, in general, have
non-trivial kernel. But, if G has cyclic Sylow p-subgroups and we restrict to
considering permutation lattices, then we show that the pairing is
non-degenerate modulo the formal kernel. Using this we can show that, for
certain groups, including dihedral groups of order 2p for p odd, the
isomorphism class of any $\mathbb{Z}_p[G]$-lattice whose extension of scalars
to $\mathbb{Q}_p$ is self-dual, is determined by its regulator constants, its
extension of scalars to $\mathbb{Q}_p$, and a cohomological invariant of
Yakovlev.
