What is the name of this oscillator topology?
I don't know the name of this oscillator, but the operation seems relatively straight forward.  


Assume current flows through L1 at the beginning and M1 is shutting off. This generates a voltage spike at Vout-, which is large enough to overcome the reverse-breakdown voltage on D2.  
This voltage spike hits the gate of M2 and turns it on for a short time. This creates a current pulse through L2 and M2.
When M2 begins to shut off, L2's inductance causes a voltage spike at Vout+. That gets sent to the gate of M1 through D1 once again by action of the reverse-breakdown voltage. 
M1 begins to generate a pulse current through it and L1 and we're back at step one.


This works because of the inductors creating voltage spikes higher than the 24V input and because the Schottky diodes are rated at 30V for reverse-breakdown. R1 and R2 are there to start up the oscillations. Without them, M1 and M2 would never turn on at all. They're also different values which supports the idea that they're just there to start the oscillations by allowing one side to temporarily override the other side.

The reason the Schottky diodes don't get fried is due to the capacitance of the FET's gate being the only path to ground. The capacitance blocks excessive current from flowing through the Schottky. Excessive current during reverse-breakdown is the mechanism of failure in them due to the heat that would be generated. That never happens here because there's never enough heat buildup to cause failure.

A basic simulation of the circuit topology shows this works. I've used different values and modeled the Schottky's as Zener's because that's more accurate use in this case because this simulator doesn't have an input for reverse-breakdown voltage:



Circuit topology simulation. Green waves are voltage while yellow is current. Vdd is only 5V.