  Resonant inelastic x-ray scattering (RIXS) has become an important tool for
studying elementary excitations in correlated materials. Here, we present a
systematic theoretical investigation of the Cu L-edge RIXS spectra of undoped
and doped cuprate two-leg spin-ladders in both the non-spin-conserving (NSC)
and spin-conserving (SC) channels. The spectra are rich and host many exotic
excitations. In the NSC-channel of the undoped case, we identify one-triplon
and bound $\textit{triplet}$ two-triplon excitations in the strong-rung
coupling limit, as well as confined spinons in the weak-rung coupling limit. In
the doped case, we observe a quasiparticle excitation formed from a bound
charge and spin-$\frac{1}{2}$ in the strong-rung coupling limit. In the
SC-channel, we also identify several new features, including bound
$\textit{singlet}$ two-triplon excitations and confined spinons in the undoped
ladders in the strong- and weak-rung coupling limits, respectively. Conversely,
in the doped case, the SC channel primarily probes both gapless and gapped
charge excitations. Finally, we revisit the available data for the ladder
compound Sr$_{14}$Cu$_{24}$O$_{41}$ in the context of our results.
