  The dimer (monomer-dimer) model deals with weighted enumeration of perfect
matchings (matchings). The monopole-dimer model is a signed variant of the
monomer-dimer model whose partition function is a determinant. In 1999, Lu and
Wu evaluated the partition function of the dimer model on two-dimensional grids
embedded on a M\"obius strip and a Klein bottle. While the partition function
of the dimer model has been known for the two-dimensional grids with different
boundary conditions, we present a similar product formula for the partition
function of the monopole-dimer model on higher dimensional cylindrical and
toroidal grid graphs. We also evaluate the same for the three-dimensional
M\"obius and Klein grid graphs and show that the formula does not generalise
for the higher dimensions. Further, we present a relation between the product
formula for the three-dimensional cylindrical and M\"obius grid.
