A planar domain \(\Omega\) is called a \(b\)-strip if each cross-section \( \Omega_x = \{y : (x,y) \in \Omega\}\) is either empty or an interval of length no greater than \(b\). The authors provide necessary and sufficient conditions on strip domains \(\Omega\) in order for them to be \(p\)-Poincaré domains, i.e. that the \(p\)-Poincaré inequality holds on \(\Omega\): A \(b\)-strip \(\Omega\) with finite area is a \(p\)-Poincaré, \(1 \leq p < \infty\), iff \(K_{p, \Omega} (w) < \infty\) for some \(w \in \Omega\). Here \(K_{p, \Omega} (w) = \sup k_p (w, \chi)^{p - 1} \text{meas} (\Omega (\chi))\) where the supremum is taken over all hyperbolic geodesics \(\chi\) with \(w \notin \chi\); the metric \(k_{p, \Omega} (u,v)\) is defined as \(\inf_\chi \int \text{dist} (z, \partial \Omega)^{- 1/(p - 1)} ds\) \((\chi\) is any arc joining \(u\) and \(v\) in \(\Omega)\) and \(\Omega (\chi )\) is the part of \(\Omega\) which does not contain \(w\). \textit{G. Pólya} [Q. Appl. Math. 6, 267-277 (1948; Zbl 0037.25301)] has shown that the Steiner symmetrization of a domain preserves an analogous inequality where the functions have zero boundary values. The authors show that the corresponding result for the Poincaré inequality (the functions have zero mean value) is false. However, the result holds for strip domains. The proofs employ certain bilipschitz methods.