I see you already have an answer based on the device physics. I'll answer that this means in a circuit.
BJT (bipolar junction transistors) do work in reverse, at least somewhat. Generally, the characteristics aren't as good, particularly the gain. However, in most cases you still get some gain with the collector and emitter swapped.
I've actually had circuits where I accidentally installed a transistor backwards, and they worked. In that case, the circuit was tolerant of the lower gain. Circuits that require closer to the minimum guaranteed gain probably wouldn't work.
How a BJT works in reverse also depends on its geometry. Planar construction, like that shown by Joel Reyes Noche, doesn't work so well in reverse. That is because it is well optimized for charges in the base to make their way to the collector. Remember that the gain is the reciprocal of those that don't make it to the collector (go to the emitter instead). If you were to flip the collector and emitter, you can visualize how the system would be less efficient, meaning not quite such a large fraction of charges in the base would make it to the collector.
In a more simple NPN or PNP sandwich, reversing the transistor won't matter as much.
I naively assumed that all transistors of the same type have essentially identical characteristics; after all, unlike for other components, there seem to be no numbers written for them on circuit diagrams.
I'm not sure what you mean by "type". NPN versus PNP, or more specific, like 2N3906 versus 2N4401. In any case, there are many many individual models of transistors. These vary between each other in gain, maximum C-E voltage, maximum C current, highest usable frequency, power dissipation capability, pinout, package, and other parameters.
Any decent schematic absolutely will show the particular transistor model, or it will be accompanied by a BOM (bill of materials) that does. Such things must be specified to be able to build the unit.
Schematics that are only intended to conceptually show a circuit, like what may be posted here on this Q&A platform, might not always specify a particular model when that's not relevant to the concept. You can't build something like that, though, without somehow picking a model for each transistor.