  We consider the numbers $\mathcal{B}_{r,s} = (\mathbf{B}+1)^r \mathbf{B}^s$
(in umbral notation $\mathbf{B}^n = \mathbf{B}_n$ with the Bernoulli numbers)
that have a well-known reciprocity relation, which is frequently found in the
literature and goes back to the 19th century. In a recent paper,
self-reciprocal Bernoulli polynomials, whose coefficients are related to these
numbers, appeared in the context of power sums and the so-called Faulhaber
polynomials. The numbers $\mathcal{B}_{r,s}$ can be recursively expressed by
iterated sums and differences, so it is not obvious that these numbers do not
vanish in general. As a main result among other properties, we show the
non-vanishing of these numbers, apart from exceptional cases. We further derive
an explicit product formula for their denominators, which follows from a von
Staudt--Clausen type relation.
