  While charge carriers can typically be designated as either electron- or
hole- type, depending on the sign of the Hall coefficient, some materials defy
this straightforward classification. Here we find that LaRh$_6$Ge$_4$ goes
beyond this dichotomy, where the Hall resistivity is electron-like for magnetic
fields along the $c$-axis but hole-like in the basal plane. Together with
first-principles calculations, we show that this direction-dependent switching
of the carrier type arises within a single band, where the special geometry
leads to charge carriers on the same Fermi surface orbiting as electrons along
some directions, but holes along others. The relationship between the Fermi
surface geometry and occurrence of a Hall sign reversal is further generalized
by considering tight-binding model calculations, which show that this type of
Fermi surface corresponds to a more robust means of realizing this phenomenon,
suggesting an important route for tailoring direction dependent properties for
advanced electronic device applications.
