The IceCube Neutrino Observatory measures cosmic-ray air showers with both its surface array IceTop and its 1.5-2.5 km deep in-ice array. IceTop measures the charge deposited by electromagnetic particles and low-energy muons. The highly abundant electromagnetic particles mainly determine the cosmic ray primary energy, while the low-energy muons, visible at the edge of the shower, add a sensitivity to the primary composition. The high-energy ($>$300 GeV) muon bundle is studied through its energy loss profile in the in-ice detector. These muons provide information about the early interactions of the cosmic ray in the atmosphere and thus to the mass of the primary particle. In this work we combine all three pieces of information. Since the yield of low- and high-energy muons differs significantly among existing hadronic interaction models, this provides a unique sensitivity to model dependent variations in cosmic-ray air shower studies.