If you stop and think about a C-R differentiator where the signal goes in the C and comes out the junction between the R and C, you will see the same spike when the triangle or square wave changes shape.
https://www.electronics-tutorials.ws/rc/rc-differentiator.htmlLook at the waveforms about 1/2 way down the page. The bottom waveform is a typical differentiator used to generate a clock pulse from a square wave. It would be cleaned up with a Schmitt Trigger perhaps but the idea is to find the edges of the square wave where dv/dt heads off toward infinity and the output waveform has the characteristic differentiator spike.
This is exactly what you have except your circuit is more elegant. But a differentiator is a differentiator and the derivative at the point of inflection or jump discontinuity is undefined.
In my view, Gibbs Phenomenon is a completely different animal and will always affect all 4 corners whereas the differentiator only looks at the 2 edges and the direction of the pulse is in the same direction as the edge transition.
Actually, GP is just a math problem. Add another harmonic and it will be reduced. Add a dozen more and it will be reduced further. Include an infinite number of terms and it will disappear. As you build up a square wave from a sine wave and its harmonics, it is the corners that are the last to fill in.
http://recordingology.com/in-the-studio/distortion/square-wave-calculations/MANY years ago, logic systems weren't always synchronous and it was quite common to use the edge of some signal to create a pulse to clock some part of the circuit.