We provide a complete characterization of pairs of full-rank lattices in
Rd which admit common connected fundamental domains of the type
N[0,1)d where N is an invertible matrix of order d. Using
our characterization, we construct several pairs of lattices of the type
(MZd,Zd) which admit a common
fundamental domain of the type N[0,1)d. Moreover, we show
that for d=2, there exists an uncountable family of pairs of lattices of the
same volume which do not admit a common connected fundamental domain of the
type $N\left[ 0,1\right) ^{2}.