  We show that the recently observed class of long-range ion-Rydberg molecules
can be divided into two families of states, which are characterised by their
unique electronic structures resulting from the ion-induced admixture of
quantum defect-split Rydberg $n$P states with different low-field seeking
high-$l$ states. We predict that in both cases these diatomic molecular states
can bind additional ground state atoms lying within the orbit of the Rydberg
electron, thereby forming charged ultralong-range Rydberg molecules (ULRM) with
binding energies similar to that of conventional non-polar ULRM. To demonstrate
this, we consider a Rydberg atom interacting with a single ground state atom
and an ion. The additional atom breaks the system's cylindrical symmetry, which
leads to mixing between states that would otherwise be decoupled. The
electronic structure is obtained using exact diagonalisation over a finite
basis and the vibrational structure is determined using the Multi-Configuration
Time-Dependent Hartree method. Due to the lobe-like structure of the electronic
density, bound trimers with both linear and nonlinear geometrical
configurations of the three nuclei are possible. The predicted trimer binding
energies and excitation series are distinct enough from those of the
ion-Rydberg dimer to be observed using current experimental techniques.
