Angular-distribution measurements have been performed on the ${\mathrm{Ne}}^{22}(p,d){\mathrm{Ne}}^{21}$ and ${\mathrm{Ne}}^{20}(d,p){\mathrm{Ne}}^{21}$ reactions at ${E}_{p}=20$ MeV and ${E}_{d}=16.4$ MeV, respectively. In the ${\mathrm{Ne}}^{22}(p,d){\mathrm{Ne}}^{21}$ reaction studies, formation of the 0-, 350-, and 3664-keV ${\mathrm{Ne}}^{21}$ levels was characterized by respective ${l}_{n}=2, 2, \mathrm{and} 1$ neutron pickup transfers. The composite ($p,d$) angular distribution observed for the experimentally unresolved 2790- and 2796-keV states is shown to require a dominant ${l}_{n}=1$ component in the associated distorted-wave Born-approximation fit, with evidence for a minor ${l}_{n}=0$ component also being present. Since the 2796-keV level possesses ${J}^{\ensuremath{\pi}}={\frac{1}{2}}^{+}$, the present ($p,d$) result implies ${J}^{\ensuremath{\pi}}={\frac{1}{2}}^{\ensuremath{-}} or {\frac{3}{2}}^{\ensuremath{-}}$ for the 2790-keV ${\mathrm{Ne}}^{21}$ level. The ${\frac{3}{2}}^{\ensuremath{-}}$, 4730-keV state did not display discernible direct-reaction characteristics in this ($p,d$) study. This was also true of the remaining ${\mathrm{Ne}}^{21}$ levels below ${E}_{x}=5$ MeV, which generally possessed \ensuremath{\sim}0.1-mb/sr cross sections attributed to second-order direct and/or compound nuclear-reaction mechanisms. The ${\mathrm{Ne}}^{20}(d,p){\mathrm{Ne}}^{21}$ results indicate ${l}_{n}=0$ transfer for the 2790-, 2796-keV ${\mathrm{Ne}}^{21}$ composite, and ${l}_{n}=1$ transfer for the 4730-keV ${\mathrm{Ne}}^{21}$ state. The 3664-keV level did not display discernible direct reaction characteristics in this ($d,p$) study. The present findings establish that the major $1p$-shell strength is located in the 2790- and 3664-keV ${\mathrm{Ne}}^{21}$ levels, whereas the major $2{p}_{\frac{3}{2}}$ particle strength is associated with the 4730-keV ${\mathrm{Ne}}^{21}$ state. These low-lying non-normal parity states are discussed within the framework of the Nilsson strong-coupling collective model.