This paper studies empty squares in arbitrary orientation among a set $P$ of $n$ points in the plane. We prove that the number of empty squares with four contact pairs is between $Ω(n)$ and $O(n^2)$, and that these bounds are tight, provided $P$ is in a certain general position. A contact pair of a square is a pair of a point $p\in P$ and a side $\ell$ of the square with $p\in \ell$. The upper bound $O(n^2)$ also applies to the number of empty squares with four contact points, while we construct a point set among which there is no square of four contact points. These combinatorial results are based on new observations on the $L_\infty$ Voronoi diagram with the axes rotated and its close connection to empty squares in arbitrary orientation. We then present an algorithm that maintains a combinatorial structure of the $L_\infty$ Voronoi diagram of $P$, while the axes of the plane continuously rotates by $90$ degrees, and simultaneously reports all empty squares with four contact pairs among $P$ in an output-sensitive way within $O(s\log n)$ time and $O(n)$ space, where $s$ denotes the number of reported squares. Several new algorithmic results are also obtained: a largest empty square among $P$ and a square annulus of minimum width or minimum area that encloses $P$ over all orientations can be computed in worst-case $O(n^2 \log n)$ time.

We prove a generalization of Pál's 1921 conjecture that if a convex shape P can be placed in any orientation inside a convex shape Q in the plane, then P can also be turned continuously through 360° inside Q. We also prove a lower bound of Omega(m n^{2}) on the number of combinatorially distinct maximal placements of a convex m-gon P in a convex n-gon Q. This matches the upper bound proven by Agarwal et al.