Bottles of Madeira wine should be stored in an upright position. A small gas exchange is supposed to keep the oxidized aroma and the contents of the bottle does not touch the cork. Otherwise the taste of the wine could suffer from the fact that the cork deteriorates faster than the wine.
Is the same true of vintage port? The Storage of Ports section in the FAQ didn't make it obvious to me.
Since Port is not oxidized as is Madeira, it should be stored on it's side. I suppose, once a Tawny or Colheita has been opened (and not before), a little oxidation can add a degree of complexity, so keeping them upright may make some sense.
Port bottles should always be matured on their side, to keep the cork moist and airtight.
Although that probably makes relatively little difference in the case of very old Tawnies and Colheitas, the T stoppers on such bottles that have been upright for many years have a habit of decaying and snapping when you try to twist them out - so best laid down.
