Oil in radiator...

 Bad water pump seals lets water out. Typical Neon head gasket failure
after 1995 is oil in rad. First. Can progress to oil outside the
engine, antifreeze in the oil, or antifreeze outside the engine.

Not possible for oil to get in the rad from the water pump.

Thanks for your input. Guess I'm looking at a bad head gasket after all.

Karl

Given the relative pressure involved the only way I can see for oil to get
into the radiator is when the system cools down and a vacuum is created in
the cooling system.

The first thing I would check would be the radiator cap to make sure it will
allow a back flow in the correct mannor.

The second assumption would be an oil leak building up on the engine where
that oil could be vacuumed into the radiator (see above).

Which rasises the question: Is it possable someone has put oil in the
radiator overflow catch basin with the intent of messing with your mind?

The "relative pressure" of the combustion chamber is much higher than
the cooling system. It's easy to find oil in the radiator with a bad
head gasket.

 You have the cause wrong, but the effect right. Or lets just say you
have it about half right. It is the fact the OIL PRESSURE is higher
than the pressure in the radiator that causes oil in the
radiator.There is no oil in the combustion chamber of a properly
operating automotive engine. The primary gasket failure on Neons AFTER
1995 is leakage from the oil gallerey to the cooling jackets and the
oil galleries to the outside of the engine. When it gets bad enough
you also gat coolant in the oil.
Prior to 1995, when the steel sandwich gasket was introduced,The
primary failure mode was leakage of combustion gasses into the cooling
system, with leakage of coolant into the combustion chamber a direct
result. When running, combustion pressure was higher than cooling
system pressure, but when you shut down a warm engine, the 14psi
cooling system pressure was higher than the combustion chamber
pressure, allowing coolant to enter the cyl.

With the new headgaskets, combustion chamber leaks are ALMOST a thing
of the past

I've always found bad headgaskets because the combustion pressure forces
coolant out of the overflow pressure release causing low coolant and
overheating. When you go to add coolant you find the muck and residue on
the cap and start-up produces white smoke from the exhaust after coolant
enters the combustion chamber on after shutdown. I've rarely seen oil in
the coolant because unless you have a very bad leak or make only short
trips, this usually boils off after driving.

 No, the reason you don't see oil in the coolant is because the
failure you are seeing is different. You see antifreeze in the oil
becuase it runs past the rings with the engine shut off and the
cooling system pressurized. This is a "combustion chamber leak" and it
also causes smelly white smoke on initial startup.

When the leak is from an oil pressure gallery, which is (relatively)
common on the later Neon, and was common on some Toyotas, for
instance, the failure does not affect the combustion chamber (at least
in the beginning), it just transfers oil into the coolant, or coolant
into the oil, depending on the pressure differential and the
characteristics of the failed gasket and location of the failure..

In these types of failures compression gasses do not get into the
coolant, overpressurizing the cooling system and blowing out coolant.