For every $\alpha\in\mathbb{R}^*$, we introduce the class of $\alpha$-embeddings as tilings of a portion of the plane by quadrilaterals such that the side-lengths of each quadrilateral $ABCD$ satisfy $AB^\alpha+CD^\alpha=AD^\alpha+BC^\alpha$. When $\alpha$ is $1$ (resp. $2$) we recover the so-called $s$-embeddings (resp. harmonic embeddings). We study existence and uniqueness properties of a local transformation of $\alpha$-embeddings (and of more general $\alpha$-realizations, where the quadrilaterals may overlap) called the cube move, which consists in flipping three quadrilaterals that meet at a vertex, while staying within the class of $\alpha$-embeddings. The special case $\alpha=1$ (resp. $\alpha=2$) is related to the star-triangle transformation for the Ising model (resp. for resistor networks). In passing, we give a new and simpler formula for the change in coupling constants for the Ising star-triangle transformation.