  Although the frequency shift and odd-even intensity modulation in high-order
harmonic generation (HHG) have both been observed for asymmetric laser-target
systems, they are typically studied as two separate phenomena. In this Letter,
we provide a comprehensive picture of these two nonlinear optical phenomena,
unifying them through a common origin - asymmetry of the laser-target system.
By tuning asymmetric laser-target systems, we discover a transition from the
harmonic frequency shift to the odd-even intensity modulation upon increasing
the duration of the driving laser pulse. Specifically, these phenomena are
observed simultaneously for laser pulses with intermediate pulse duration. For
numerical evidence, we solve the time-dependent Schr\"{o}dinger equation, while
insight into the underlying physics is obtained from a simplified analytically
tractable model. Understanding the asymmetric characteristics reflected in the
HHG as provided is crucial for retrieving laser-target information, sampling
external fields, and probing molecular dynamics.
