A Poincaré-covariant $\mathrm{quark}+\mathrm{diquark}$ Faddeev equation is used to develop insights into the structure of the four lightest $(I,J^P=\frac{3}{2},{\frac{3}{2}}^{\pm })$ baryon multiplets. While these systems can contain isovector-axialvector and isovector-vector diquarks, one may neglect the latter and still arrive at a reliable description. The $(\frac{3}{2},{\frac{3}{2}}^{+})$ states are the simpler systems, with features that bear some resemblance to quark model pictures, e.g., their most prominent rest-frame orbital angular momentum component is $S$-wave and the $\Delta \left(1600\right){\frac{3}{2}}^{+}$ may reasonably be viewed as a radial excitation of the $\Delta \left(1232\right){\frac{3}{2}}^{+}$. The $(\frac{3}{2},{\frac{3}{2}}^{-})$ states are more complex: the $\Delta \left(1940\right){\frac{3}{2}}^{-}$ expresses little of the character of a radial excitation of the $\Delta \left(1700\right){\frac{3}{2}}^{-}$; and while the rest-frame wave function of the latter is predominantly $P$-wave, the leading piece in the $\Delta \left(1940\right){\frac{3}{2}}^{-}$ wave function is $S$-wave, in conflict with quark model expectations. Experiments that can test these predictions, such as large momentum transfer resonance electroexcitation, may shed light on the nature of emergent hadron mass.