MHz conductivity, torque magnetometer and magnetization measurements are reported on single crystals of CeOs4Sb12 and NdOs4Sb12 using temperatures down to 0.5 K and magnetic fields of up to 60 tesla. The field-orientation dependence of the de Haas-van Alphen and Shubnikov-de Haas oscillations is deduced by rotating the samples about the [010] and [011] directions. The results indicate that NdOs4Sb12 has a similar Fermi surface topology to that of the unusual superconductor PrOs4Sb12, but with significantly smaller effective masses, supporting the importance of local phonon modes in contributing to the low-temperature heat capacity of NdOs4Sb12. By contrast, CeOs4Sb12 undergoes a field-induced transition from an unusual semimetal into a high-field, hightemperature state characterized by a single, almost spherical Fermi-surface section. The behavior of the phase boundary and comparisons with models of the bandstructure lead us to propose that the field-induced phase transition in CeOs4Sb12 is similar in origin to the well-known α − γ transition in Ce and its alloys.