  We exhibit a finite basis M for a certain variety $\mathbf{V}$ of medial
groupoids. The set M consists of the medial law (xy)(zt)=(xz)(yt) and five
other identities involving four variables. The variety $\mathbf{V}$ is
generated by the four groupoids $\pm x\pm y$ on the integers. Since
$\mathbf{V}$ is a very natural variety, proving it to be finitely based should
be of interest.
  In an earlier paper, we made a conjecture which implies that $\mathbf{V}$ is
finitely based. In this paper, we show that $\mathbf{V}$ is finitely based by
proving that M is a basis. Based on our proof, we think that our conjecture
will be difficult to prove.
  We used four medial groupoids to define $\mathbf{V}$. We also present a
finite basis for the variety generated by any proper subset of these four
groupoids. In an earlier paper with R. Padmanabhan, we gave the corresponding
finite bases when the constant zero is allowed.
