  The combined information from cosmic ray air showers that trigger both the
surface and underground parts of the IceCube Neutrino Observatory allows the
reconstruction of both the energy and mass of the primary particle through the
knee region of the energy spectrum and above. The properties of high-energy
muon bundles, created early in the formation of extensive air showers and
capable of penetrating deep into the ice, are related to the primary energy and
composition. New methods for reconstructing the direction and
composition-sensitive properties of muon bundles are shown. Based on a
likelihood minimization procedure using IceCube signals, and accounting for
photon propagation, ice properties, and the energy loss processes of muons in
ice, the muon bundle energy loss is reconstructed. The results of the
high-energy muon bundle reconstruction in the deep ice and the reconstruction
of the lateral distribution of low energy particles in the surface detector can
be combined to study primary composition and energy. The performance and
composition sensitivity for both simulated and experimental data are discussed.
