  Regular black holes are often geodesically incomplete when their extensions
to negative values of the radial coordinate are considered. Here, we propose to
use the Simpson-Visser method of regularising a singular spacetime, and apply
it to a regular solution that is geodesically incomplete, to construct a
geodesically complete regular solution. Our method is generic, and can be used
to cure geodesic incompleteness in any spherically symmetric static regular
solution, so that the resulting solution is symmetric in the radial coordinate.
As an example, we illustrate this procedure using a regular black hole solution
with an asymptotic Minkowski core. We study the structure of the resulting
metric, and show that it can represent a wormhole or a regular black hole with
a single or double horizon per side of the throat. Further, we construct a
source Lagrangian for which the geodesically complete spacetime is an exact
solution of the Einstein equations, and show that this consists of a phantom
scalar field and a nonlinear electromagnetic field. Finally, gravitational
lensing properties of the geodesically complete spacetime are briefly studied.
