The properties of the subspaces of the many-body Hilbert space, which are associated with the use of the generator coordinate method in connection with one-parameter and with two-conjugate-parameter families of generator states, are examined in detail. These families are obtained by letting unitary displacement operators, having as generators canonical operators $\stackrel{^}{P}$ and $\stackrel{^}{Q}$, defined in the many-body Hilbert space, act on a reference state. We show that natural orthonormal base vectors in each case are immediately related to Peierls-Yoccoz and Peierls-Thouless projections, respectively. Through the formal consideration of a canonical transformation to collective, $\stackrel{^}{P}$ and $\stackrel{^}{Q}$, and intrinsic degrees of freedom, we discuss in detail the properties of the generator-coordinate-method subspaces with respect to the kinematical separation of these degrees of freedom. An application is made, using the ideas developed in this paper, (a) to translations, (b) to illustration of the qualitative understanding of the content of existing generator-coordinate-method calculations of giant resonances in light nuclei, and (c) to the definition of appropriate asymptotic states in current generator-coordinate-method descriptions of scattering.