Republican presidential nominee Donald Trump, right, with his running mate, Indiana Gov. Mike Pence. (Evan Vucci/Associated Press)

DONALD TRUMP’S Wednesday staff shake-up suggests the billionaire candidate will double down on the ugliness that helped him win the primaries. If so, it will sharpen the dilemma for Republicans who have hitched their fortunes to the Trump campaign — beginning with his more conventional running mate, Indiana Gov. Mike Pence.

Mr. Pence has already indicated that he will diverge from Mr. Trump on at least one important matter: political transparency. The GOP vice-presidential nominee indicated last weekend that he would release his tax returns. Mr. Pence said there would be nothing earth-shattering in the financial forms he will turn over. Yet the fact of disclosure itself would be remarkable. It would mean that the GOP vice-presidential candidate, the lesser partner on a major-party ticket, would have submitted himself to a higher standard of scrutiny than the man whose character voters most need to understand.

Mr. Trump has said that he cannot release his tax returns because the Internal Revenue Service is auditing several years of tax forms, and that older returns are “linked” to those under audit. But there is no reason he cannot release returns still under review — along with any “linked” returns. Mr. Trump swore to the U.S. government that his returns were accurate. The public has a legitimate interest in seeing what he signed his name to.

Revealing tax returns signals respect for the voters and their right to review the conduct of those asking to be elected to the world’s most powerful office. Every major-party presidential nominee since 1976 has released his or her tax returns. Hillary Clinton, Mr. Trump’s Democratic opponent, has released decades of them. Her running mate, Sen. Tim Kaine (Va.), released his promptly after his nomination. Mitt Romney, the previous GOP presidential nominee and a wealthy businessman in his own right, resisted but ultimately complied with public pressure to release his tax forms.

Even Mr. Romney, meanwhile, had a record in public office that voters could review. Mr. Trump has never held public office, and most of his private dealings are shrouded in secrecy. The Trump Organization, his business, is a closely held private company. If any candidate has a special responsibility to offer voters financial information, it is Mr. Trump.

There are a variety of speculations about what Mr. Trump might be hiding. Perhaps, as documents from decades ago suggest, he has paid little or no taxes. Maybe he earns less than he has led voters to expect. It could be that he gives little to charity. It could be a combination of all of these. What’s clear is that Mr. Trump’s spurning of essential transparency is politically out of bounds. His running mate apparently understands as much.