In case of an electoral college tie, can a regular House vote prevent the "en bloc" House vote from happening?

Question

Should neither candidate for president get to 270 votes in the electoral college, the twelfth amendment to the constitution specifies that the House of Representatives "shall choose immediately" a new president, but with one vote for each state.

Should one party have serious issues with that candidate, and assuming they control the majority in the "normal" House, could they block this process? What happens if they decide to not schedule this vote in the House's agenda, despite the constitution specifically obligating them to immediately do this vote?

My first guess would have been that the "en bloc" House could set the agenda for this one vote themselves. The issue is that "a quorum for [the] purpose [of this vote] shall consist of a member or members from two-thirds of the states", so if both parties have at least one member in two-thirds of the states, both parties can form this quorum, and validly vote in their preferred president without members of the opposite party present. As this is absurd, clearly the location and time of this vote should be decided by another authority instead. Is this other authority the "normal" House?

Answer 1

Like many interesting questions like this, there is a Congressional Research Service report that addresses this.

Voting “Immediately” and “by Ballot”

The 12th Amendment next provides that the House “shall choose immediately, by ballot ... the President.” Most observers agree that the first part of this clause—“immediately”—requires that the House must literally proceed to the contingent election without any delay. It should also be noted that the rules adopted for contingent election in 1825 required the House to “ballot for a President, without interruption by other business, until a President be chosen.”

So, given the "immediately" requirement, no other business - including changing the timing or rules on voting - can interrupt the vote.

That said, in the modern day and age, everyone knows what the results of the Electoral College are before Congress officially opens the votes and tallies them. That provides a window of time whereby the rules for the 12th Amendment vote can be set by the new House between when they first sit and when they have the joint session to open the electoral votes, in anticipation of having to make that decision.

For the logical followup question of "what if they just reschedule the opening of the votes?", see this answer. (The Senate also gets to reschedule if they want, since it has to be a joint session.)