What are the spiral marks in the center of the engines?
The benefit of these white markings inside the engine is twofold:


The ground crew can tell if the engine is running or not.
When an airplane is at the gate before departure, there are several personal attending the airplane. They would be loading luggage, refueling, removing waste products, loading food, etc. etc.. Not all of these people are in constant contact with the pilots and they can come very close to the engines. Although there are several precautionary measures taken before the pilots turn on the engines, it is a very clear visual indication by looking at the spirals to tell if the engines are running or not. Rotor or prop blades sometimes have their tips painted for the same reason.
The birds keep away.
During flight, when the engine is running, birds see the spinning spiral as opposed to just a black hole, and hence they stay away from it. It is like an anti-bird signal because if there was nothing in the middle it just looks like a cave or black void that birds tend to fly toward. Even though it looks like a blur to us, birds can see motion better. Wikipedia mentions this also:


  ... it appears as a white circle which discourages birds from flying into the engine.



Rolls-Royce brochure (PDF) quoted the following (identified by user BowlOfRed in comment below):


  Our aerospace engines have swirls painted onto their spinners in order
  to indicate when the engine is rotating while on the ground. In flight
  these swirls flicker as the engine rotates at high speed, scaring
  birds and allowing them to fly clear of the engine.