It's like this, but rearranged slightly.
That's not rearranged slightly, that's a different circuit!
Well, no. Move the winding from supply to base, to supply to emitter. Flip the phase, so B-E voltage is the same, and adjust turns ratio to account for the difference in C-E voltage.
(Grid-cathode coupled blocking oscillators were very common in tube TV deflection oscillators, probably because it's easy to injection-lock that topology.)
A circuit may look different while behaving identically. Any circuit transformation that results in the same equations (or close enough given the operating conditions) has this property. The simplest transformation is simply shifting ground, which immediately leads to a variety of oscillator circuits, for example (a few of which are named). Swapping series elements (e.g., supplies and reactive components -- see above) leads to topologically different but functionally identical circuits, as does splitting or combining components (Colpitts has a split capacitor and single inductor, others use a single cap and tapped inductors or multiple windings).
Tim