We investigate the dispersive waves (DWs) emitted from shaped pulses with spectral Heaviside step phases (HSPs). The spectrally HSP-modulated pulse exhibits a unique double-peak structure, where the intensity and separation of the twin peaks are determined by the modulation depth and frequency detuning. By tailoring the parameters of the HSP suitably, we can control the DW emission with regard to resonant frequency and conversion efficiency. As the intensity ratio or relative separation of neighboring peaks is elaborately chosen, the DW emission can be effectively boosted, or a solitonic cage can be constructed for realizing temporal reflections and refractions associated with spectral broadening and multi-peak spectra of the output DWs. These findings offer a straightforward and efficient approach for controlling the DW emission, which is highly relevant to the advancement of supercontinuum generation and wavelength conversion technology.