The answer is Yes, of course, a warrior can grasp his own sword by the blade, and the old fighting manuals show this technique clearly (half-sword technique).
There are also instances where it is recommended to grasp the opponent's sword.
Here is good treatise on the topic of half-sword technique, when it's used & why.
From Medieval Combat (a translation & description of Talhoffer's Fechbuch), we read: "The half=sword techniques were originally devised for fighting armoured opponents. The sword is gripped on the blade with the left hand and used like a short spear or bayonet to slice or stab into the armpits, groin, face, throat and joints."
Even though the technique was devised for armoured fighting, it could certainly be used for fighting without armour. Apropos to the question of injury to the swordsman's hand, note that medieval guantlets did not fully encase the hand and fingers:
The swordsman still had to wear a (thin) leathern glove underneath. This does not invalidate the answer, because the leather itself is relatively thin and offers little protection from stabs and slices. Thin leather = less bunching & greater range of motion; thick leather = bunching & less range of motion. You still need to take care in order not to slice your hand through the glove!
Zweihander Techniques Demonstration