The presence of net circular dichroism in the photoionization of nonchiral homonuclear molecules has been put in evidence recently through the measurement of molecular-frame photoelectron angular distributions in dissociative photoionization of ${\text{H}}_{2}$ [Dowek et al., Phys. Rev. Lett. 104, 233003 (2010)]. In this work we present a detailed study of circular dichroism in the photoelectron angular distributions of ${\text{H}}_{2}$ and ${\text{D}}_{2}$ molecules, oriented perpendicularly to the propagation vector of the circularly polarized light, at different photon energies (20, 27, and 32.5 eV). Circular dichroism in the angular distributions at 20 and to a large extent 27 eV exhibits the usual pattern in which inversion symmetry is preserved. In contrast, at 32.5 eV, the inversion symmetry breaks down, which eventually leads to total circular dichroism after integration over the polar emission angle. Time-dependent ab initio calculations support and explain the observed results for ${\text{H}}_{2}$ in terms of quantum interferences between direct photoionization and delayed autoionization from the ${\mathcal{Q}}_{1}$ and ${\mathcal{Q}}_{2}$ doubly excited states into ionic states $(1s{\ensuremath{\sigma}}_{g}$ and $2p{\ensuremath{\sigma}}_{u})$ of different inversion symmetry. Nevertheless, for ${\text{D}}_{2}$ at 32.5 eV, there is a particular case where theory and experiment disagree in the magnitude of the symmetry breaking: when ${\text{D}}^{+}$ ions are produced with an energy of around 5 eV. This reflects the subleties associated to such simple molecules when exposed to this fine scrutiny.