Is a recoil start (hand propping) allowed without it being referenced in the aircraft manual? In a Cessna 150, for example.
You didn't say exactly what "allowed" means, e.g. do you mean approved by the manufacturer, or legal in a certain country? But if you're asking is it legal in the US (under FAA regulations), then the answer is yes.
AOPA has a nice article on exactly this question, called Hand propping: A legal primer. It says:
There is no specific FAA regulation that applies to hand propping an airplane, either to prohibit it or to direct how it is to be done
The FAA contends that hand propping is a two-person operation and has expressed this view in the Airplane Flying Handbook (FAA-H-8083-3A) under the section titled “Hand propping.” Of course, this publication is not regulatory, but the NTSB was surely influenced by it in a 1983 legal decision. In that case, the FAA sought to suspend a pilot’s certificate for being careless or reckless when, while attempting to start a VariEze experimental aircraft, it “got away” and ran into a parked aircraft.
The NTSB concluded that the pilot had violated 14 CFR 91.13 (careless and reckless operation):
The board affirmed the administrative law judge’s finding that there had been a 91.10 (now 91.13) violation
That means that hand propping itself isn't illegal, but doing it wrongly and against the FAA's general procedures could get you in trouble:
There have been at least two previously issued NTSB (full board) decisions and one subsequent decision that refer to these generally accepted procedures and precautions for hand propping. The precedent has been set. So, hand proppers beware; if you fail to follow proper precautions and the airplane gets away, the FAA might pursue action against you for being careless or reckless.
Unless an operation is prohibited by regulation, the laws of physics, or the limitations section of the POH/AFM, it's not illegal. It might be stupid, but it's not illegal.
There is an important distinction to be made here. The manufacturer can state that hand propping "is not recommended" but that is not a binding statement and hand propping is still permitted...though not recommended. Simply because a procedure is not recommended does not mean it's prohibited. Now, if the manufacturer places a limitation of "no hand propping" in the limitations section (the real, formal section of the POH called "Limitations"), then the procedure is not authorized at all.
I know of no airplane with this limitation because it's not really a problem. I think our good friends physics and Darwin take care of that for us. I've never seen a person hand prop a GTSIO-520 (physics) but if someone manages to hand prop a large piston engine I doubt we'll ever hear about it (Darwin).
Smaller piston engines are actually easy to hand prop. Hand propping of O-170s (Continental A series) and O-190s (Continental C series) is actually quite common because many of the airplanes equipped with these engines lack electrical systems to power starters. The C-150 uses a Lycoming O-200 which is only 10 cubic inches larger than the Continental O-190s. The O-200 is rarely hand propped, however, because the aircraft that use it generally have electrical systems and starters. Why hand prop when you can just use a key?
When I was flying skydivers we had occasion to hand prop a Cessna 182. That's almost a 500 cubic inch engine, and hand propping it is definitely NOT in the manual. But all reciprocating aircraft engines have impulse-coupled magnetos, so you don't really have to turn them quickly to get them to fire. When you move the propeller through the sweet spot, the mag will unwind and cause a spark. The coupling makes a ticking noise that you can hear. If there's fuel in the jug when the magneto fires it will kick, and that's usually enough to start it.