Basic Electrical – How Can a Circuit Lack a Neutral?

What completes the circuit? My understanding is that hot provides charged particles, while the neutral gives any remaining charged particles a path back to ground.

Nope, that's not how it works. Both the hot wire and the neutral wire are capable of providing charged particles—which is to say, electrons. And when an appliance draws electrons from one wire, it pushes the same number of electrons into the other wire, not just "any remaining" ones. The amount of charge (which is to say, the number of charged particles) going in equals the amount of charge going out.

But if the charge going in equals the charge going out, how does all this motion of charge accomplish anything? If I may quote myself,

The answer is, the electric charge doesn't simply glide effortlessly through your light bulb. The electric company forcefully pushes charge in through the hot wire and forcefully pulls charge out through the neutral wire. Then the process reverses direction; the electric company forcefully pushes charge in through the neutral wire and forcefully pulls it out through the hot wire.

Now, this process works in pretty much the same way whether you have a hot wire and a neutral wire, or two hot wires out of phase. The difference is what happens with the electric potential.

Electric potential is, essentially, the amount of force with which the power supply is trying to push or pull electrons. If the electric potential in a wire is high (by which I mean highly positive), that means that there's a force trying to pull electrons into the wire. If the potential is low (by which I mean lowly positive, or highly negative), there's a force trying to push electrons out of the wire.

With a hot wire and a neutral wire, you have one wire whose electric potential is constantly changing (high, low, high, low, ...) and one whose electric potential is staying the same (medium, medium, medium, medium, ...). And that works just fine. While the hot wire's potential is high, the appliance allows electrons to pass through itself from neutral to hot. And while the hot wire's potential is low, the appliance allows electrons to pass from hot to neutral. In both cases, the appliance is obtaining energy from that forceful push or pull of charge.

With two hot wires out of phase, you have one wire where the potential is going "high, low, high, low, ..." and one where it's going "low, high, low, high, ..." And that still works, just the same. The appliance allows electrons to pass through itself, from low potential to high potential, and so it obtains energy, just like in the paragraph above.

What would happen if you had two hot wires in phase, where both wires are high at the same time, and then both wires are low at the same time? Not much. The appliance can't obtain any energy if both wires are pushing equally hard at the same time, or pulling equally hard at the same time. There needs to be a difference in electric potential in order for the appliance to work.