What is a "pole-marked" door?

Solution 1:

When I was a lad, barber shops always had one of these outside.

Older versions were a simple pole painted in red and white stripes.

The attached also gives you the, slightly gruesome, history of the barber's pole - which date from when barbers were surgeons and blood-letters as well as hairdressers and shavers.

I'm not sure what "cycle through" refers to. In Britain we certainly never cycled into a barbers shop. Perhaps in America they had sort of drive-through barbers!

Solution 2:

A pole-marked door refers to an old-fashioned barber shop's sign. See one here: https://en.wikipedia.org/wiki/Barber%27s_pole

Usually, the "poles" (AKA signs) were hung outside the shop. Marked here means: a sign was hung to the side the door indicating it was a barbershop.

Marked just means indicating. It is not so clearly expressed because the signs were usually separate from the door or hung off a column that was part of an actual building. Near the door. But not really "marking" the door.

They are red and white and blue. Their is not a consensus of the colors: the US flag or two types of blood (venous and arterial) and a bandage.

Here is a picture of one that is actually more of a "pole": http://www.waymarking.com/waymarks/WMGG75_3_Aces_Barber_Shop_New_York_NY