I was in a 172M club that had a bad first order shake appear after an
engine overhaul. It turned out the prop had to be indexed to 45
degrees @ TDC to get rid of it. The orientation requirements are in
the Cessna manual, but there is no explanation.

As an ME with some background in dynamics, this is the only reason I
can see for this:

All 4 cylinder horizontally opposed engines with offset (in contrast to
directly opposite) cylinders will have a second harmonic yaw motion
about their CG due to the short connecting rods. Without a double
crank speed counterbalance shaft system, there is no way to eliminate
this. This causes a yaw vibration or swinging of the engine at 80 Hz
at 2400 rpm for example. I think it is also the reason the vibration
environment on the front of a Lycoming engine is so severe.

A two blade propeller has to present the same yawing moment of inertia
in response to the 80 Hz engine swing. It can do so only if it is
phased such that it is oriented 45 degrees to the plane of the crank
throws.

If it is phased in the plane of the crank throws, the engine swinging
will present different yaw inertias to alternating second harmonic
inputs ( i. e. every other one), causing a first order response of the
system. This first order response will appear as a shake at crankshaft
frequency.

This would not be a problem we had three blade propellers, or 6
cylinder engines.

Anybody want to challenge this? I'd be interested in further
discussions on this topic