By Rae Wee and Harry Robertson

SINGAPORE/LONDON (Reuters) -The dollar rebounded sharply on Tuesday after plunging hours earlier as Donald Trump's first day back in the U.S. presidency brought mixed messaging on tariffs and highlighted investors' sensitivity to headlines about trade policy.

U.S. stock futures rose after bouncing around on Monday as traders digested Trump's statements on economic policy and trade levies, while European equities eked out small gains.

The Mexican peso and Canadian dollar tumbled on Monday night - reversing sharp gains from earlier in the day - after Trump said he was mulling imposing 25% tariffs on the neighbouring countries as soon as Feb. 1.

That sent the Mexican peso sliding 1.3% against the U.S. dollar while the Canadian dollar tumbled to a five-year low of $0.689.

Around 11 hours earlier the U.S. dollar had plunged against its peers after a presidential memo, first reported by the Wall Street Journal, said the administration would probe trade issues but stop short of day-one tariffs.

The dollar index, which measures the currency against six peers, was last up 0.64% at 108.69 on Tuesday, although failed to make back the 1.2% it lost on Monday in its biggest daily fall since November 2023.

"The first few hours of the Trump administration have underscored that the policy environment will be dynamic once again and markets should brace for volatility," said Charu Chanana, Saxo's chief investment strategist.

"Clearly, the markets celebrated too soon with tariff threats missing at the outset in Trump's inaugural speech."

The euro was last down 0.62% at $1.2248, after rallying 1.4% on Monday amid the relief about a delay in tariffs. Sterling was 0.67% lower after jumping 1.3% the previous day.

Trump also said he wanted to reverse the U.S. trade deficit with the European Union, either with tariffs or more energy exports.

Europe's continent-wide STOXX 600 index was 0.16% higher but Germany's DAX was down 0.1%.

U.S. stock futures rallied after struggling in early trading, leaving Nasdaq futures 0.56% higher and S&P 500 futures up 0.49%.

European carmakers slipped around 0.4% on Tuesday after Trump revoked a 2021 executive order signed by his predecessor that sought to ensure half of all new vehicles sold in the United States by 2030 were electric.

TRUMP TEMPEST

The dollar has risen around 5% since Trump won the Nov. 5 election, partly because the U.S. economy has remained strong and partly as investors have braced for wide-ranging tariffs that would likely hurt America's trading partners.

Many investors had expected early action on tariffs, hence the big moves sparked by the memo indicating no immediate action.

U.S. Treasury yields were down 4 basis points on Tuesday at 4.57%, after reopening for trading after a public holiday.

The yields, which set the tone for borrowing costs around the world, are nonetheless up almost a percentage point since the Federal Reserve slashed rates by 50 bps in September, reflecting an economy that has continued to grow strongly.

Chinese stocks were volatile but ended higher as Trump largely steered clear of definitive threats against the country's exports, with the CSI 300 index up 0.1%.

Trump warned he could impose tariffs on China if Beijing failed to approve a U.S. deal to be a half-owner of short-video app TikTok.

"At some point, we are quite certain that Trump will start to move on the tariff measures," Khoon Goh, head of Asia research at ANZ, said of tariffs more broadly.

"The fact that he hasn't addressed this on day one doesn't mean that it is off the agenda. It is definitely firmly on the agenda, it's just that we have to wait and see what shape or form he takes."

Elsewhere, Trump's new crypto token gave up some of its strong gains on Tuesday to fall 20% to $35.27, after having soared to more than $10 billion in market value at the start of the week.

(Reporting by Rae Wee and Ankur Banerjee in Singapore, Jiaxing Li in Hong Kong, and Harry Robertson in London; Editing by Sonali Paul and Jamie Freed)