Initial value representations are constructed to avoid the search for trajectories that are only defined in semiclassical approximations by their boundary conditions. We show how to incorporate these procedures within the full Weyl representation, so that quantum expectation values are given by phase space integrals over the evolving Wigner function. Spurious semiclassical singularities at caustics are cancelled, even though there is no increase in the number of trajectories, as compared to usual semiclassical formulae. The whole construction remains exact in the case of quadratic Hamiltonians. The evolution of (density) operators depends on either a forward and backward trajectory, given by an initial value, or else on a pair of trajectories, propagating backwards from their final value. The latter option reduces numerical errors in the computation of trajectories. The general scheme also leads to analogous algorithms for evolving the quantum fidelity, which can be approximated perturbatively with a single trajectory, reducing to the 'dephasing representation' for small times. The theory is developed within a generalized 'Maslov method', based on semiclassical Fourier transforms, in order to avoid singularities in the limit of small times.