Square Wave with Random Edge Shifting
By separately changing both the leading edge and the trailing edge of each pulse, you have converted the square wave into what can be viewed as an asymmetric double-edge pulse-width-modulated (DEPWM) signal. Analysis of the Fourier spectra of various types of deterministic PWM signals (including the type mentioned in the previous sentence) can be found in the paper Z. Song and D. V. Sarwate, "The Frequency Spectrum of Pulse Width Modulated Signals", Signal Processing, vol. 83, pp. 2227-2258, 2003. (The article is available behind the publisher's paywall but there are some free copies floating around the web).  The results show that the spectra of PWM signals do have the spikes shown in the OP's diagram just as the square wave spectra do, but there is additional clutter between the peaks. That is, the PWM Fourier spectrum is not a Fourier series like the square wave spectrum is; PWM spectra are of a continuous-time signal that contains periodic components (giving those spikes) as well as non-periodic components giving the clutter.  The details (and the nuances) are too messy to transcribe here; interested readers should read the original.