I don't really think Ichigo's Hollow side is his "fighting spirit"...it's just his Hollow side. By all accounts, it shouldn't be there; Ichigo is a freak fusion of Shinigami and Hollow, which aren't meant to exist alongside one another (all Urahara's fault, of course, even if I do believe it was accidental; it was the Shattered Shaft incident that did it to him). The reason why the Hollow side seems to represent power and stuff could be because the fusion makes him even stronger than he could have been normally, due to the fact that such a combination breaks the natural "power caps" that normal Shinigami or Hollows have. But the Hollow doesn't represent anything about Ichigo; it's just there, really.
And I suppose you could say that the Hollow mask is a like a Zanpakutou; the Zanpakutou represents Ichigo's Shinigami abilities, while the mask represents the presence of a Hollow deep within him. Like Shinigami and Hollows themselves, the mask and the sword are like two sides of the same coin.