We wash our hands twice in the Passover seder—the first time without saying any blessing, and the second time with the conventional brachah. I wrote last year that the reason commonly giving for omitting a blessing at the first washing—so that someone will ask the reason—seems a bit silly.
I suggested then that the reason might be that the rabbis who compiled the haggadah might not have been sure that it was really a mitzvah to wash at that point. As a rule, if there is uncertainty about whether something really is a mitzvah or not, we perform the action but omit the blessing. In this case, just before the karpas, there could be doubt because we're not ready to eat bread (matzah) or the full meal; we're only eating a rather small amount of a vegetable.
Here's another thought. Perhaps we wash the first time, not to fulfill the mitzvah of netilat yadayim, but because of the requirement of our ancient priests to wash before performing certain sacred rituals. After the destruction of the Second Temple, the rabbis taught us that the family dining table would assume some of the functions of the altar, albeit without sacrifices. The Passover seder itself can be seen as a replacement for the Pesach sacrifice, which is specifically represented on the seder plate by the shankbone (zeroa).
If the seder is the replacement for the sacrifice, it follows that we ourselves are the stand-ins for the priests. Thus, it would make sense for us to wash at the beginning of the seder as a remembrance of the requirement for the priests to wash.