This study formulates numerical and analytical approaches to the self-equilibrium problem of novel units of tensegrity metamaterials composed of class θ = 1 tensegrity prisms. The freestanding configurations of the examined structures are determined for varying geometries, and it is shown that such configurations exhibit a large number of infinitesimal mechanisms. The latter can be effectively stabilized by applying self-equilibrated systems of internal forces induced by cable prestretching. The equilibrium equations of class θ = 1 tensegrity prisms are studied for varying values of two aspect parameters, and local solutions to the self-equilibrium problem are determined by recourse to Newton–Raphson iterations. Such a numerical approach to the form-finding problem can be easily generalized to arbitrary tensegrity systems. An analytical approach is also proposed for the class θ = 1 units analyzed in the present work. The potential of such structures for development of novel mechanical metamaterials is discussed, in the light of recent findings concerned with structural lattices alternating lumped masses and tensegrity units.