No "checklists"! Hard-built interlocks only.
There is no such thing as "making sure your main breaker and 2-pole breakers are turned off". You must not do generator interlocking via a checklist or procedure.
There are only three legitimate ways to switch from mains to backup power:
- a generator-style interlock or transfer switch
- "permanently" rewire the panel or circuits so they hardwired to power off the alternate source only, and no longer are able to power from mains. When power returns, "pemanently" wire them back.
- switch the loads via cord-and-plug connection.
The cheapest and simplest way to implement a generator interlock is, if an interlock kit isn't made for your main panel, to get a no-lug subpanel with an inexpensive interlock (Murray or QO) and have two backfeeding breakers back to back interlocked, and then put the loads you want to switch into this panel.
Go ahead and feed both poles, except...
Once you have done this, go for it. Go ahead and split your 120V supply so it feeds both legs of the interlocked 240V "generator" breaker.
... except Multi-Wire Branch Circuits
However, this will be very bad for multi-wire branch circuits (MWBC). These are circuits which share a neutral. The hot wires must be on opposing poles so the neutral only handles differential currents. Putting them on the same pole will mean neutral handles the sum of both hots' currents, and that can overload it.
How do you avert this? Ideally, you put your to-be-gen-powered circuits in a separate subpanel (again, the cheap way to provide that gen interlock)... and simply don't migrate MWBCs to that panel.
Or go on a crusade to eliminate all multi-wire branch circuits,
At the very least, you identify in advance your MWBCs, by going through your panel, finding them, and placing them on 2-pole breakers.
Notably, a "make turning off MWBCs part of your checklist" apparently violates the "no checklists" rule. However the main reason for that reg is so you don't kill linemen, who are innocent outsiders. I am not sure if Code has taken a stance on this entirely local consequence.
What will happen if you power a 240V appliance?
A 240V appliance (water heater, A/C): Nothing. Every conductor in the appliance will be at the same potential, so no electrons will move. This will happen to be 120V from safety ground, but who cares.
A 120/240V appliance (dryer, range): The 120V parts of the appliance will power up (tumbler, timer, clock, oven light). The 240V parts, see above. So for instance your dryer will work, but only on "fluff/no heat".