In this paper we construct new wormhole solutions of Lovelock theories in vacuum, when the coupling constants are such that all the maximally symmetric solutions coincide, extending to arbitrary dimensions wormhole solutions previously known in the Chern-Simons case. Like the latter, the wormholes are characterized by an integration constant rho 0 that controls the contribution to the energy content from one of the boundaries. Then, we study the effects of the constant rho 0 on the spectrum of a massive, (non)minimally coupled scalar probes, with Dirichlet boundary conditions at both asymptotic regions. As a result, a deformed Breitenlohner-Freedman bound emerges, which is sensitive to the value of rho 0. The scalar spectra are numerically obtained in detail in dimension five, and in such dimension we also present a new family of wormhole geometries for the Einstein-Gauss-Bonnet theory with a unique vacuum. The new geometries are constructed via a double analytic continuation of a wormhole previously reported in the literature, but now the constant rho 0 appears in the centrifugal terms of the equations for the geodesic and scalar probes. The mass of these configurations vanishes nontrivially, since the contributions to the mass integral from each boundary are nonvanishing, but only differ in sign, providing a new example of a spacetime having "mass without mass."