I thought “reconciliation” was a procedure used in the Senate, not the House. Am I mistaken?
Not exactly. Reconciliation is a process created in the Congressional Budget Act of 1974 that does affect both the House and the Senate - but, it probably affects the Senate more because it eliminates the filibuster on these very narrowly defined budget-related bills.
The so-called "nuclear option" (a phrase coined by Trent Lott in the first part of the last decade), describes a parliamentary maneuver that dates back to before the 1830s that would abolish, at least temporarily, the filibuster. Such a maneuver could theoretically be used on any kind of bill in the Senate, not just budget bills.
My understanding is that reconciliation is the process by which bills with different language from each chamber are conformed for final passage -- the conference committee. There are some specifics laid out, as OldDeckHand notes, in budget bills, Congressional Budget Act of 1974, due to the necessity of funding the functions of government, can't have it grind to a standstill because the House and Senate can't agree on this or that wording. Because this legislation they are foisting is not budget/appropriations, it has to go through normal legislative process ... unless they cheat, as it seems the Dems are attempting to do here.