Here is a more likely scenario that could take place under existing law.
Let's say Trump and Clinton get the Republican and Democrat nominations, Bernie runs as a third party candidate, and Bernie manages to get just enough electoral votes to prevent Trump or Clinton from getting a majority. Meanwhile, the Democrats retake the Senate.
The House would then pick the next President (voting by state, so each state gets one vote). The House can only pick from the top three candidates who got votes as President. Would the House pick Trump or Clinton? All Clinton needs is enough RINOs to join with the Democrats and give her a majority of the states.
One thing working in our favor is that each state gets one vote, regardless of population or number of congresscritters, and most of the Democrats are concentrated in a few large states. So all of those flyover states with low populations, like Montana and North Dakota, have the same number of votes as New York and California.
The Senate then gets to pick the Vice President, and is limited to picking from the top two VP candidates. So even if the Republican controlled House picks Trump as President, the Democrat controlled Senate could pick Clinton's VP candidate as the Vice President.
kennedy, you are correct, a law adopted pursuant to Section 4 of the 20th Amendment would have no bearing on the 12th Amendment’s provision that if no one gets a majority of EVs that the House, voting by delegation, shall elect the president. There is no hidden agenda at play here.
And you also are correct that if the election gets thrown to the House that RINOs could bandy together with Democrats to elect Hillary, although only if the GOP suffers big House losses would the Democrats control 26 delegations even with the help of Hillary-supporting RINOs.