What an interesting idea to consider. Beowulf is an epic Anglo-Saxon poem/story which chronicles the exploits of Beowulf. One of the three adversaries whom Beowulf battles and defeats is Grendel. Because they are opposite in nearly every way, perhaps Grendel can best be seen as Beowulf's mirror image rather than his doppelganger.
We do not learn very much about the monster Grendel, we are told that he "nursed a hard grudge" (Seamus Heaney translation). This bitterness causes him to strike out in anger and evilness throughout Hrothgar's kingdom. He goes on a killing rampage. We never hear him speak or see him interact in any way with society. He prefers the dark and the dank.
Beowulf, on the other hand, has nothing but noble motives for his actions. He comes to Hrothgar's rescue with the lofty goal of killing the marauder. He makes lofty speeches and enjoys the company of his fellow soldiers and Hrothgar's court. He does not have any obligation to fight for Hrothgar, but he is a man of honor and justice.
Beowulf has heard that Grendel fights without weapons, so he plans to do the same. The battle becomes a literal arm-wrestling match, each of them with an extraordinary strength in their arms (hands). The visual of the two men locked in a grip of death is suggestive of a mirror image. Though the image ends as Beowulf is the victor and Grendel slithers away to his marshy fen to die, the two are clearly opposites--thus the mirror image.