Phase or polarity for North America residential 240V
The general situation in North America (I also was not aware of the NYC ConEdison situation mentioned by nanofarad in another answer) is that homes are provided with as much as \$200\:\text{A}\$ service (my situation, for example) with what is called &quot;split-phase 240&quot;.
The key thing to remember is &quot;split-phase,&quot; as that describes the situation well. If you removed the neutral between the two &quot;hots,&quot; L1 and L2, then you'd just have &quot;single-phase 240.&quot; But the neutral &quot;splits&quot; this single-phase situation and provides two &quot;hots&quot; with half the voltage and opposite phase.
Separately from the above, it's also common practice in North America, to tie the neutral to the literal ground either by using copper stakes driven deep into the ground near the panel service entry to a home or else by using what's called a Ufer ground (worth reading the history of that, as it involves WW II and storing explosives in concrete bunkers and the threat of a &quot;spark&quot; triggering unwanted events there.) A Ufer ground basically just uses the concrete foundation, in contact with the Earth ground, as a grounding for neutral. The requirement is that there is at least a 20' continuous length of 1/2&quot; rebar in the concrete bed itself to act as a conductor. (NEC also permits the alternative of 20' of bare #4 copper instead of the rebar.) This grounding of Neutral is also done at points where higher AC voltages are converted through a power transformer into the residential service voltages and also at substations and elsewhere throughout the system.
This grounding of Neutral is largely because of safety considerations as a floating residential service could, due to local thunderstorms and other reasons, rise to dangerous voltage differences between all of the home conductors and the local Earth ground. By strongly tying the Neutral to the local Earth ground itself, none of the conductors (neutral, L1, or L2) should be able to rise too much above the local voltage of the ground you walk upon (or any conducting fixture, such as plumbing, that you might touch.)

Is there such a thing as phase or polarity between these three wires?

From the perspective of L1, L2 looks like a \$240\:\text{V}_\text{RMS}\$ AC &quot;hot.&quot; From the perspective of L2, L1 looks like a \$240\:\text{V}_\text{RMS}\$ AC &quot;hot.&quot; From the perspective of Neutral, L1 looks like a \$120\:\text{V}_\text{RMS}\$ AC &quot;hot&quot; and L2 also looks like a \$120\:\text{V}_\text{RMS}\$ AC &quot;hot.&quot; But from the perspective of Neutral, L1 look like it is always of opposite going phase as compared with L2.

How is hot to hot 240 different than hot to neutral 120 in this
respect?

From a perspective of powering a device that only expects to operate from single-phase AC, there's no practical difference except for the difference in supply voltage. So you can operate a toaster using Neutral and L1, or else Neutral and L2, and at the same time operate an oven using L1 and L2 (and not Neutral, except as a chassis safety connection.) Single-phase \$120\:\text{V}_\text{RMS}\$ AC devices don't care if they operate from L1 or from L2. (No practical difference.)
But wiring within the residence is usually arranged (by some managed thinking) so that about half of the single-phase \$120\:\text{V}_\text{RMS}\$ AC device loads will be attached to L1 and the other half attached to L2.
Imagine the simplest situation where you have two toasters, one hooked to L1 and one hooked to L2 and both, of course, with their other ends hooked to Neutral. The residential service will mostly power the two toasters through each other (from L1, through toaster 1, to toaster 2, and then to L2) and won't invoke any currents in the Neutral leg back to the residential power transformer. This is the optimal situation. But, of course, one of the toasters will pop up (and turn off) earlier than the other and when that happens, then Neutral back to the residential power transformer will carry current.
There are standards that apply. ANSI C84.1, &quot;American National Standard for Electric Power Systems and Equipment—Voltage Ratings (60 Hertz)&quot;, covers many of the details. National Electrical Code (NEC, in the US) covers additional details and is more concerned with risks related to electrical fire (and shock) safety than details about the expected service voltages. A new version of ANSI C84.1 is scheduled for publication in the month of June, 2020, with an eye towards higher allowable voltage ranges and/or voltage rises in dedicated circuits for renewable energy resources (to coordinate with IEEE 1547 and the National Electrical Code.)
A voltage table reflecting ANSI C84.1 can be found here:

You can see the working draft of the newer ANSI C84.1 here.