  We show that eigenvalues of the Robin Laplacian with a positive boundary
parameter $\alpha$ on rectangles and unions of rectangtes satisfy
P\'{o}lya-type inequalities, albeit with an exponent smaller than that of the
corresponding Weyl asympotics for a fixed domain. We determine the optimal
exponents in either case, showing that they are different in the two
situations. Our approach to proving these results includes a characterisation
of the corresponding extremal domains for the $k$th eigenvalue in regions of
the $(k,\alpha)$-plane.
