An investigation of the ($p,d$) reaction on $N=Z$ nuclei in the $2s\ensuremath{-}1d$ shell has been made to obtain spectroscopic information and to study the ${l}_{n}=2$ $J$-dependence for the ($p,d$) reaction. The experiments were performed with $^{24}\mathrm{Mg}$, $^{28}\mathrm{Si}$, $^{32}\mathrm{S}$, $^{36}\mathrm{Ar}$, and $^{40}\mathrm{Ca}$ as target nuclei, and virtually all of the $2s\ensuremath{-}1d$ shell hole strength was observed. Deuteron angular distributions for strongly excited levels in $^{23}\mathrm{Mg}$, $^{27}\mathrm{Si}$, $^{31}\mathrm{S}$, $^{35}\mathrm{Ar}$, and $^{39}\mathrm{Ca}$ were measured for laboratory angles from 10\ifmmode^\circ\else\textdegree\fi{} to 155\ifmmode^\circ\else\textdegree\fi{}, and spectroscopic factors were obtained using distorted-wave Born-approximation (DWBA) calculations. The $J$ dependence for the pickup of an ${l}_{n}=2$ neutron appears mostly in the forward angles of the angular distributions and seems to follow systematic trends through the $2s\ensuremath{-}1d$ shell, thus suggesting spin assignments for levels in $^{31}\mathrm{S}$, $^{35}\mathrm{Ar}$, and $^{39}\mathrm{Ca}$. Appreciable configuration mixing is found to exist in the ground-state wave functions of all the nuclei investigated. Of particular interest are the ${l}_{n}=1$ levels excited in the $^{24}\mathrm{Mg}(p,d)^{23}\mathrm{Mg}$ and $^{28}\mathrm{Si}(p,d)^{27}\mathrm{Si}$ reactions, which could arise from the removal of either $1p$- or $2p$-shell neutrons. The ground states of $^{36}\mathrm{Ar}$ and $^{40}\mathrm{Ca}$ are observed to contain appreciable mixing with the ${f}_{\frac{7}{2}}$ shell, and evidence exists for a small ${[2p]}^{2}$ admixture in the $^{40}\mathrm{Ca}$ ground state. The level orders of the residual nuclei and the DWBA spectroscopic factors are discussed in terms of the strong-coupling rotational model and Nilsson-model wave functions. Evidence for strong rotational band mixing is apparent in many cases.