A major unsolved problem in proteid dynamics is the connection between the amino acid sequence and the propensity to fold to a unique, ordered structure. Many globular proteins fold to native states with great regularity in their tertiary structure, reminiscent of symmetry. This paper considers the relation between sequence, regular folded structure, and formal group theoretical symmetry in lattice protein models. in computer simulations of model proteins consisting of II chain of hydrophobic (H) and polar (P) units on a lattice [K. Yue and K. A. Dill, Proc. Natl. Acad. Sci. USA 92, 146 (1995)] it was observed that some chains fold to regular, "symmetric" structures. These structures have a small number of native states, i.e., the conformations of minimum energy have low degeneracy in this model, in which the energy depends only on the number of H-H contacts. The present work is concerned with the connection between the symmetry properties of the H-P sequence and those of the folded lattice protein. The meaning of symmetry in lattice models is discussed in terms of mathematical groups. The concepts of the lattice symmetry, the hydrophobic lattice core and its geometrical symmetry, and the core sequence and its permutation symmetry, are defined and related. With the help of a simple rescaling of the energy, the folded lattice protein is seen to have the symmetry of a microcrystal of H and P "atoms" arranged on a lattice. Geometrical symmetry of the folded structure is related to formal permutation symmetry properties of the H-P sequence, in order to establish the connection between sequence and fold. How these relations among symmetry, structure, and sequence might relate to folding in real proteins, including the design of sequences to fold into desired structures, is discussed.