  We find a new class of theories of massive gravity with five propagating
degrees of freedom where only rotations are preserved. Our results are based on
a non-perturbative and background-independent Hamiltonian analysis. In these
theories the weak field approximation is well behaved and the static
gravitational potential is typically screened \`a la Yukawa at large distances,
while at short distances no vDVZ discontinuity is found and there is no need to
rely on nonlinear effects to pass the solar system tests. The effective field
theory analysis shows that the ultraviolet cutoff is (m M_PL)^1/2 ~ 1/\mu m,
the highest possible. Thus, these theories can be studied in weak-field regime
at all the phenomenologically interesting scales, and are candidates for a
calculable large-distance modified gravity.
