The spectra of quasiperiodically modulated quantum Ising chains are investigated. For moderate values of the modulation strength a unified approach, based on the lifting of the degeneracy between the left- and right-moving eigenmodes is proposed. The local scaling behaviour reflecting partial restoration of translational invariance is established. Both the conformally invariant scaling for the lower modes and the multi-fractal scaling for the higher ones are described in terms of the same quantity. A simple but accurate calculational scheme is constructed, where the rearrangement in the level spectra is due to the interaction between left- and right-moving modes of the periodic system as induced by the quasiperiodic modulation.