The inelastic scattering of 18.6-MeV polarized protons has been studied for the following nuclei: $^{48}\mathrm{Ti}$, $^{50}\mathrm{Ti}$, $^{52}\mathrm{Cr}$, $^{54}\mathrm{Fe}$, $^{56}\mathrm{Fe}$, $^{58}\mathrm{Ni}$, $^{62}\mathrm{Ni}$, $^{64}\mathrm{Ni}$, and $^{63}\mathrm{Cu}$. The targets $^{52}\mathrm{Cr}$, $^{60}\mathrm{Ni}$, and $^{62}\mathrm{Ni}$ have been investigated at 16.5 MeV. The measured asymmetries for strong $l=2$ transitions tend to fall into two categories, distinguished by the magnitude of the asymmetries at 30\ifmmode^\circ\else\textdegree\fi{} and 90\ifmmode^\circ\else\textdegree\fi{}. Of those $l=2$ transitions studied, only those to the first ${2}^{+}$ state of the 28-neutron nuclei seem to show large asymmetries at these angles. Strong $l=3$ and $l=4$ transitions also reveal several interesting variation. When the entire optical potential is deformed, coupled-channels and distorted-wave Born-approximation calculations predict the "small" $l=2$ asymmetries reasonably well, but the fits to the large asymmetries are unsatisfactory. Phenomenological calculations in the spirit of the microscopic model indicate that the predicted asymmetry is sensitive to the form factor. No important differences between $S=0$ and $S=1$ predictions were found. Simple calculations with form factors arising from single-particle transitions predict asymmetries which closely resemble those predicted with a real collective-model form factor.