An interacting cylindrical boson model, describing rotating cylindrical intrinsic states, is utilized to study the bandhead states of the rotational bands in rare-earth nuclei. The three fundamental quasispin (\ensuremath{\Lambda}) bosons, ${p}_{s}$ (\ensuremath{\Lambda}=0) and ${p}_{\ensuremath{\lambda}}$ (\ensuremath{\Lambda}=(1/2), are introduced and used to solve Hamiltonians with cylindrically broken SU(3) symmetry. This approach, simply incorporated into the interacting boson model SU(3) rotational limit, gives some new results in four areas of interest: changes in order of the strict SU(3) bandhead sequences, explanations for observed low-lying ${0}^{+}$ rotational bands not accounted for in the pure SU(3) limit, certain bandhead mixing effects, and analytical expressions for E2 transition rates between bandhead states. We also obtain further insights into the interacting boson model SU(4) sp\ifmmode\bar\else\textasciimacron\fi{} boson model of Chen and Arima. We illustrate with new results for $^{156}\mathrm{Gd}$, $^{168}\mathrm{Yb}$, and $^{168}\mathrm{a}$ sequence of well-studied nuclei having increasing interacting boson approximation valence boson number.