California Gov. Jerry Brown  wouldn’t top anyone’s list of tax-and-spend liberals. Famous for his frugality, he ushered in his latest stint as governor by replacing the sleek boardroom table in his new office with a hard, wooden picnic table.

But there aren’t enough picnic tables in the world for the savings to close California’s $27 billion deficit. Now Mr. Brown finds himself locked in a showdown with Republican legislators over whether the budget will be balanced with a new round of tax increases or deeper cuts in spending.

Mr. Brown is pushing to place a proposal on the June ballot that would raise an estimated $12 billion by extending some temporary tax increases set to expire in July and revive a tax increase that expired in December. The quickest way to do this would be with a two-thirds vote by both houses of the legislature.

But he needs two more Republican votes in both the Assembly and the Senate, and GOP leaders have made it clear that those votes will come at a price. They released Friday a seven-page list of policy demands, including reductions in public-employee pensions and a state spending cap, that they want to see enacted in exchange for their support for the tax increases.