In this paper, we present a unified microscopic description of low-lying collective states in even-even rare-earth nuclei using the angular-momentum-projected number-conserved BCS (PNBCS) method within a shell-model space. Systematic calculations for seven isotopic chains from Xe to Dy naturally generate both vibrational and rotational band structures, reproduce level energies, $B\left(E2\right)$ transition strengths, gradual shape evolution, sharp shape-phase transitions, and produce results consistent with the empirical $N_p{N}_n$ systematics as well as the $Z=64$ subshell effects. The analysis highlights the importance of the interplay between pairing correlations and proton-neutron quadrupole-quadrupole correlations. Pairing correlations are crucial for reproducing spectra and the observed moments of inertia, but have little influence on nuclear shapes or $B\left(E2\right)$ strengths. Proton-neutron quadrupole-quadrupole correlations, together with the $Z=64$ subshell closure and orbital occupations, govern the evolution or transition of collectivity and explain the saturation of the $B\left(E2\right)$ strengths in well-deformed nuclei relative to the pseudo- and quasi-SU(3) schemes. Discrepancies in seniority-like nuclei indicate the importance of broken-pair configurations, which are beyond the present framework. Predictions have been made for neutron-rich isotopes with levels or $B\left(E2\right)$ strengths that have yet to be measured.

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• Collective levels
• Electromagnetic transitions
• Energy levels
• Nuclear shapes and moments
• Nuclear structure & decays
• Phase transitions
• Shell model
• r process
• 150 ≤ A ≤ 189
• 90 ≤ A ≤ 149