The decays ${}^{110}\mathrm{Rh}$ and ${}^{112}\mathrm{Rh}$ have been investigated using on-line mass separation with the ion-guide technique. Extended decay schemes have been constructed for both the low- and high-spin states in ${}^{112}\mathrm{Rh}$. Mixing ratios for the collective transitions from the ${2}_{2}^{+}$ and ${3}_{1}^{+}$ states in ${}^{112}\mathrm{Pd}$ have been measured by $\ensuremath{\gamma}\ensuremath{-}\ensuremath{\gamma}$ angular correlation. The presence of two sets of ${0}^{+}$ and ${2}^{+}$ states in the 1.1\char21{}1.4 MeV range suggests the existence of an intruder band, the energy of which is the lowest in ${}^{110}\mathrm{Pd}$ with two neutrons fewer than the midshell. The quasiparticle levels at 2195 and 2755 keV are assigned $I=4$ and $I=5$, respectively. The corresponding levels in ${}^{110}\mathrm{Pd}$ are a new level at 2261 keV and the 2805 keV level. Systematics of log$\mathrm{ft}$ values and excitation energies of these quasiparticle levels is remarkably smooth. The strong $\ensuremath{\beta}$ feeding to the $I=5$ quasiparticle state can be regarded as similar to the main branch in the decay of odd Rh isotopes, while a neutron is a spectator.