Well, the answer to the part about spark ignition timing is that this engine does indeed make twice the number of sparks than the number of compression strokes.
This is not the only engine that does. Just about all engines that have a magneto ignition timed to the crankshaft do this. It is called "wasted spark ignition" because the magneto fires every time the flywheel magnet passes the magneto (coil). The useful spark occurs on the compression stroke, and the wasted spark occurs on the exhaust stroke.
What are you basing the statement "It's getting fuel" on?
You might be getting fuel as far as the carburetor inlet but that does not mean fuel is actually being processed through the carburetor and fed to the engine.
Shoot a couple of shots of spray carburetor into the throat of the carb and then try to start the engine.
If the engine runs for a few seconds on the carb cleaner (call it "alternative fuel"), then the carb is not feeding fuel to the engine.
Look at the image below to help you find the model and type numbers of your engine.
The surface you will find the numbers on is the sheet metal cooling shroud of the engine.
Try the carb spray, and then post back with the model & type numbers and we will proceed from there.

{{gwi:325617}}

I have already tried the gas down the carb throat, and ether too. Won't fire at all. On full choke it does spit a little gas out the intake while cranking (air cleaner off). Pull each plug and turn it over and they spark frequently across the gap. It seems even more frequent than every TDC but it may be. Compression is good. The engine is low hours, btw. It is a replacement engine that was installed a few years ago and the tractor is used lightly.
There are no numbers on the shroud that I can find. Even used a mirror on a stick to search all around the engine. The shrouds are clean.

Look down the throat of the carburetor with the choke wide open (choke control in off position) and the throttle control on the dash set to highest speed, use a flashlight if needed.
Can you see the throttle plate (butterfly) standing up vertically in the full fuel, wide open throttle position, or is it closed?
If you find the throttle plate is closed, look for a broken spring on the governor and carb linkage. Your carb and governor linkage should have 2 springs.
Refer to the image below (image was lableled for a different thread) for example of spring locations.

What you stated about "getting fuel" was confirmed by your pouring some gas into the carb throat, so the problem is not fuel.

{{gwi:335225}}

Observing the throttle plate when I move the control, I see the butterfly valve going from closed to about half open. The upper half of the control range doesn't open it any farther. It seems to hit a spring stop, which I suppose may be the governor control. FYI, my governor mechanism looks completely different than your picture.
Thanks for the help.

Loosen the screw clamp holding the choke cable and adjust so choke comply opens, re-tighten clamp.

Walt Conner

As you stated, it's a Twin II (Type numbers 1100 and higher). I have no photos of the II version, but the excerpted image below depicts the II governor springs.
When you have the speed control (dash throttle control) set at highest speed with the engine not running, the governor spring should pull the throttle shaft/plate to the wide open position.
But.........the engine should be at least attempting to run with the throttle plate opening even half way. And a cold engine should try to start with fuel and choke applied.
I am tempted to point my finger at the magneto because I know from experience that sometimes a coil or a magneto will not spark across the plug gap with the plug installed and under the influence of compression, but if you take the plug out of the cylinder and ground it on the engine block it will show spark. Briggs ignition tester (special tool) takes this phenomenon into account by permitting tyhe spark plug to remain in the cylinder (exposed to compression) while you watch for spark/arc in the electrodes of the ignition spark tester.
So, what you are seeing as spark outside the engine might not be making spark under the influence of compression.
E-mail me for a PDF manual covering all the Opposed L-head Twins.

{{gwi:340116}}

Thanks Mownie, I misread.

Walt Conner