By a 1941 result of Ph. M. Whitman, the free lattice {{,mathrm{FL},}}(3) on three generators includes a sublattice S that is isomorphic to the lattice {{,mathrm{FL},}}(omega )={{,mathrm{FL},}}(aleph _0) generated freely by denumerably many elements. The first author has recently “symmetrized” this classical result by constructing a sublattice Scong {{,mathrm{FL},}}(omega ) of {{,mathrm{FL},}}(3) such that S is selfdually positioned in {{,mathrm{FL},}}(3) in the sense that it is invariant under the natural dual automorphism of {{,mathrm{FL},}}(3) that keeps each of the three free generators fixed. Now we move to the furthest in terms of symmetry by constructing a selfdually positioned sublattice Scong {{,mathrm{FL},}}(omega ) of {{,mathrm{FL},}}(3) such that every element of S is fixed by all automorphisms of {{,mathrm{FL},}}(3). That is, in our terminology, we embed {{,mathrm{FL},}}(omega ) into {{,mathrm{FL},}}(3) in a totally symmetric way. Our main result determines all pairs (kappa ,lambda ) of cardinals greater than 2 such that {{,mathrm{FL},}}(kappa ) is embeddable into {{,mathrm{FL},}}(lambda ) in a totally symmetric way. Also, we relax the stipulations on Scong {{,mathrm{FL},}}(kappa ) by requiring only that S is closed with respect to the automorphisms of {{,mathrm{FL},}}(lambda ), or S is selfdually positioned and closed with respect to the automorphisms; we determine the corresponding pairs (kappa ,lambda ) even in these two cases. We reaffirm some of our calculations with a computer program developed by the first author. This program is for the word problem of free lattices, it runs under Windows, and it is freely available.