The hallmark of Weyl semimetals is the existence of open constant-energy contours on their surface - the so-called Fermi arcs - connecting Weyl points. Here, we show that, for time-reversal symmetric realizations of Weyl semimetals, these Fermi arcs, in many cases, coexist with closed Fermi pockets originating from surface Dirac cones pinned to time-reversal invariant momenta. The existence of Fermi pockets is required for certain Fermi-arc connectivities due to additional restrictions imposed by the six Z2 topological invariants characterizing a generic time-reversal invariant Weyl semimetal. We show that a change of the Fermi-arc connectivity generally leads to a different topology of the surface Fermi surface and identify the half-Heusler compound LaPtBi under in-plane compressive strain as a material that realizes this surface Lifshitz transition. We also discuss universal features of this coexistence in quasiparticle interference spectra.