  We study Hermitian metrics with a Gauduchon connection being "K\"ahler-like",
namely, satisfying the same symmetries for curvature as the Levi-Civita and
Chern connections. In particular, we investigate $6$-dimensional solvmanifolds
with invariant complex structures with trivial canonical bundle and with
invariant Hermitian metrics. The results for this case give evidence for two
conjectures that are expected to hold in more generality: first, if the
Strominger-Bismut connection is K\"ahler-like, then the metric is pluriclosed;
second, if another Gauduchon connection, different from Chern or
Strominger-Bismut, is K\"ahler-like, then the metric is K\"ahler. As a further
motivation, we show that the K\"ahler-like condition for the Levi-Civita
connection assures that the Ricci flow preserves the Hermitian condition along
analytic solutions.
