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Donald Trump hasn't shown an ounce of concern for any of the millions of Americans being hurt by his demanded federal government shutdown, but the moment he learned his actions would cost him the most lavish and flattering television appearance of the year, a State of the Union address, he suddenly found something to care about. Now comes the news that he's threatening to just show up and give his speech anyway, regardless of whether House Speaker Nancy Pelosi and the rest of the House gives him permission, because of course he is. It's the most Trump thing ever.
So, can he do that? No. He can't do it, and it's not worth even getting bothered over. His staff absolutely knows this as well, which makes the current outburst all the more pathetic, and there are a lot of individual rules and details involved, but from a logistical standpoint there's no mechanism by which the White House can commandeer House chambers to hold a Joint Session of Congress on their own say-so.
Without both House and Senate resolutions to do so, there will not be a Joint Session for Trump to address. Senators will not be there. The House may be in session or in recess; if Pelosi wanted to send Congress home on Jan. 29 she could do so. And if House Democrats wanted to invite another possible speaker, perhaps a recently retired president with his own thoughts on the current situation, to give a speech on the day and hour Trump was planning to show up, they could do that as well.
What the presidency does give Donald Trump is the right to enter the House while it is in session and wander around the floor a bit. He is free to do so; he cannot, however, simply commandeer the dais and start shouting things out like an unusually angry improv comic. It is at this point that the Sergeant at Arms could possibly decide to cart the nice man off, if push came to shove, but a far more likely outcome would be for House leadership to put the House in recess, ordering the microphones turned off. C-SPAN coverage is, as we were rather frequently reminded of during the Paul Ryan era, fully at the discretion of the House Speaker; the cameras would immediately be turned off as well. And possibly the lights, so Trump had better bring along a flashlight.
Obviously, all of this gets extremely silly very quickly. To be sure, Donald Trump could smuggle in his smartphone and video himself giving a speech to nobody in a possibly dark and empty room. Given that the whole point of Trump's fit is the denial, by Pelosi, of the pageantry of the State of the Union, it seems vanishingly unlikely he would willingly put himself in a position that would be mocked, relentlessly, for the rest of his term.
Trump could instead give a speech to the Senate, providing Senate Leader Mitch McConnell could muster the required 60 votes to invite him. Trump could give a speech somewhere else, to a rally crowd or from the Oval Office or from the bumper cars of an amusement park. But none of these things are a State of the Union address to Congress. They are just speeches. There is indeed no requirement that Trump give a speech at all: The Constitution only demands the president deliver a report to Congress, always done in writing before the newer and more camera-friendly tradition of speechmaking began. (Trump has shown little patience for any other part of the Constitution, and one might half suspect him to refuse to submit a written version just so he can brag to his base that he is offending that Constitution in yet another novel way.)
From a tactical standpoint, all Donald Trump has to do to get the televised, applause-line-packed State of the Union speech he so desperately wants is to wait. He could simply not be a child about it. Wait until government reopens, something that he himself has near-absolute control over, upon which Speaker Pelosi has indicated she will re-invite Trump, approve the necessary Joint Session, and Trump gets his evening of much-needed adulation. If Trump does anything else, that invitation likely goes away; Pelosi will note that Trump has already given his little speech, albeit in the Oval Office or in a bumper car, and that will be that.
That Trump can't do that, and in fact seems insistent on staging a faux-State of the Union that would scuttle all chances for a real one, is yet another indication of the man's shallow compulsiveness. He can't stomach the notion that he does not have the power to do exactly what he wants, and would rather lash out in foolish-looking ways than just put on his grown-up pants and deal with the momentary setback. So be it. House Democrats aren't going to be interested in keeping Donald Trump from looking foolish. If he wants to make a scene on the House floor or in the White House, you can be sure Nancy Pelosi won't be holding him back.
Update: