In the traditional method for Saya, the burnished mune or spine should be the only part of the sword which touches the inside of the saya. The tip and polished areas of the sword should not come into contact with the insides of the saya. This is accomplished by the saya being carved correctly, working in conjunction with a properly made Habaki. The Habaki tapers from tip to back and the widest part at the back fits tightly into the koiguchi or mouth of the scabbard. It is this fit which holds the blade in place within the saya. Also, authentic Japanese saya do have the small pocket carved into the tip to collect excess oil.
In antique swords, collected for their artistic aesthetics, and which have undergone expensive polish, it is extremely important, no an absolute must, that the sword does not come in contact with the interior of the saya. Contact could cause scratches or damage to your new and very expensive polish.
Swords used for Iaido or Iaijustu, or functional swords used for cutting still fit and rest within the saya the same way. The difference being that these working blades can and do make contact along the insides of their saya. Drawing the sword requires a rotation of the sword from the carried position of edge up, spine down. It is rotated outwards between 45° - 90° and when drawn the sides of the blades may come into contact with the saya. Drawing and cutting inevitably causes Hikekizu or surface scratches, which is why these types of swords do not undergo expensive art polishes.
Hope this helps.
