A two-sided Faulhaber-like formula involving Bernoulli polynomials

We give a new identity involving Bernoulli polynomials and combinatorial numbers. This provides, in particular, a Faulhaber-like formula for sums of the form $1^m(n-1{)}^m+2^m(n-2{)}^m+\cdots +(n-1{)}^m{1}^m$ for positive integers $m$ and $n$.