Why a worm gear cannot be driven backwards
Self-locking, and when that is a feature rather than a limitation.
A worm drive cannot usually be driven backwards because the worm's thread is cut at an angle shallow enough that the friction between it and the wheel it meshes with is enough, on its own, to resist any force the wheel tries to push back through it, exactly the same friction-driven locking that keeps a shallow, well-cut wedge from being pushed back out once it has been driven home.
Why friction, not geometry, holds this one shut
A worm looks like a screw thread meshing with a gear's teeth, and it behaves like one in exactly the way that matters here, since driving the worm turns the wheel easily through ordinary meshing action, while trying to turn the wheel and have that motion drive the worm backwards runs directly into the same friction that lets any shallow screw thread resist being pushed straight through under an axial load alone. Whether that friction is enough to lock the mechanism depends on a direct contest between the worm's own thread angle and the friction available at the contact between the worm and the wheel, and a worm cut with a shallow enough thread angle relative to that friction locks solidly no matter how hard the wheel tries to back-drive it, while a worm cut with a steeper thread angle can be back-driven with enough force, exactly the way a coarser, steeper screw thread is easier to strip loose under axial pressure than a fine one.
The door-wedge comparison
A doorstop wedge driven firmly under a door with a light tap holds the door open reliably against a determined push from the door itself, the door's own weight and momentum pressing straight down onto the wedge's sloped face without ever managing to slide it back out, because the wedge's shallow angle means the friction along its sloped contact surface is more than enough to resist being pushed backwards by a force arriving from that direction. Tap the wedge itself sideways, along the direction it was originally driven in, and it slides free easily, since that direction of force is exactly the one the wedge's friction was never fighting against in the first place. A worm gear locks against the wheel for precisely this same reason, resisting force arriving from the wheel's own direction while turning freely and easily the moment the worm itself, the equivalent of tapping the wedge from its own natural direction, is what is actually driving the motion.
Why this self-locking is deliberately useful
A worm gear's inability to be back-driven turns out to be exactly the property wanted in a great many mechanisms, a hoist or a winch that must hold a suspended load steady the instant the motor stops, a steering mechanism that should not be pushed backwards by road shocks transmitted up through the wheels, since the worm itself supplies a genuine mechanical lock with no separate brake, ratchet or clutch needed to keep the load from creeping back the moment driving force stops being applied. This is a fundamentally different kind of self-locking from the toggle mechanism covered earlier in this set, which holds shut through pure geometry past a dead point rather than through friction, and a worm gear's version of the same useful property, staying put without continued force, comes instead from a friction relationship that has to be deliberately engineered into the thread angle rather than from any dead-centre geometry at all.
One figure worth keeping in mind
A worm gear's lead angle, the thread's own steepness, typically has to stay below a specific threshold set by the friction actually present at the contact surface for the mechanism to lock reliably, and a worm cut only slightly past that threshold, whether from an intentionally steeper design or from wear gradually changing the effective friction over time, can begin to back-drive under load in a way it never did when new. This is exactly why a worm gear mechanism relied upon to hold a load safely deserves periodic checking rather than a one-time assumption that its self-locking property, verified once when new, will simply continue holding indefinitely regardless of wear or lubrication changes over its service life.
What follows from this
Choosing a worm gear specifically for its self-locking property means treating that property as a genuine design requirement worth verifying deliberately, checking the actual thread angle against the actual friction expected in service, rather than assuming self-locking behaviour comes automatically with anything built around a worm and a wheel. A worm gear cut with too shallow a lead angle purely for a different reason, compactness or a particular ratio, can end up self-locking as an unplanned side effect, which is harmless where it happens to help and a genuine problem where the mechanism was actually meant to be back-driveable and now silently is not.
Where this stops being true
Not every worm and wheel pairing is self-locking, since a worm cut with a sufficiently steep lead angle, or running with unusually low friction from a well-chosen lubricant or a low-friction material pairing, can be back-driven under enough force despite superficially looking like every other worm gear on the shelf. Self-locking is a genuine, calculable property of the specific angle and friction involved, not an automatic guarantee that comes bundled with the worm-and-wheel shape on sight alone, and a mechanism designed around the assumption of self-locking deserves an actual check of that specific angle and friction rather than a glance at the general shape of the parts involved.
The lubricant choice deserves particular attention here, since a fresh, low-friction lubricant intended purely to reduce wear and improve efficiency can, as a side effect, lower the friction enough to push a marginally self-locking worm gear over into one that can be back-driven after all, a genuine trade between running efficiency and holding safety that is easy to overlook when a lubricant is chosen purely to make the mechanism run more smoothly.