How can a circuit not have a neutral?
The two sides are out of phase with each other. When one side is high, the other is low, and vice versa.
This is a 240 V outlet.
tl; dr: You should install a NEMA 14-50 (4-wire: L1/L2, Neutral and Ground) instead of NEMA 6-50 for best flexibility. NEMA 14-50 is the standard 'dryer' plug now, and it's what a car charger uses too.
By the way, if you're messing with this, please consult an electrician. The fact that you're asking this at all makes me all kinds of worried.
Now, to the question. A 240V-only device has no need for a neutral since the load is connected across both ‘hot’ legs. So the current is always in balance, thus no need for a neutral to carry any imbalance. You will sometimes find this kind of wiring on high-power appliances like water heaters, ovens and fixed A/C units: two hots L1 and L2, and a Ground.
Plug-ins like dryers and electric ranges? That's a confusing story - sometimes there's just Neutral and sometimes Neutral and Ground. At the risk of drifting into DIY-land... more below.
3-wire vs. 4-wire 240V
Historically in the U.S. there have been two different types of high-current plug-in 240V appliance wiring: "3-wire" and "4-wire".
3-wire 240V has L1, L2 (hot) and Neutral. Appliances like dryers and ranges were allowed to connect Neutral to frame ground, so effectively the Neutral was serving as safety ground and carrying imbalance current if the appliance also used 120V.
Sound sketchy? It is. Even when things are working properly, when an appliance uses both 240V and 120V, the Neutral will have a small AC voltage on it due to the IR drop of the wire carrying the imbalance current. But when things go bad - say, for some reason Neutral opens up due to a flaky connection - you will have full 120V on the chassis. Not safe at all.
To remedy this fault, newer (since '96) 240V wiring uses the 4-wire system with L1, L2 (Hot), Neutral and Ground. Neutral is optional, only if the appliance needs a split 120 feed (say, for a dual-wattage heating element or a controller that runs on 120V), but it always gets a Ground. This enforces the idea that safety ground never carries current except when there's a fault, which isn't possible with the 3-wire L1/L2/Neutral setup.
More about the two 240V wiring types here: https://fredsappliance.com/service/3-prong-vs-4-prong-dryer-outlets-whats-the-difference/
And here: https://www.jadelearning.com/3-wire-cords-on-modern-4-wire-appliances/
Either way, at the panel the two hot legs L1/L2 are referred back to Neutral, as well as to safety ground. L1/L2 will have a duplex panel breaker, one breaker for each leg. The duplex breaker ensures that a hot-to-ground, hot-to-neutral or hot-to-hot fault in either leg will trip both breakers.
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.