The authors compute the monodromy groups of certain families of bivariate systems of partial differential equations of hypergeometric type and investigate their properties. They solve the closely related problem of describing all holomonic bivariate hypergeometric systems in the sense of Horn whose solution space splits into a direct sum of one-dimensional monodromy-invariant subspaces for almost all values of the parameters. Special attention is paid to invariant subspaces of Puiseux polynomial solutions.