By Jan Strupczewski

BRUSSELS (Reuters) - Fighting tax avoidance should be a cornerstone of a push by EU authorities for greater equality as they seek a response to a rise in populism across the continent, the head of euro zone finance ministers said on Tuesday.

Large multinational corporations must be made to help meet society's costs, Jeroen Dijsselbloem said, following last week's European Commission ruling that iPhone maker Apple <AAPL.O> should pay 13 billion euros of back taxes to Ireland.

"Every individual or company should contribute its fair share," he told a gala dinner organized by think-tank Bruegel.

"That means ...increasing tax compliance. This is a fundamental issue to fairness."

He also said bank owners and investors should foot the bill if their institution failed, rather than taxpayers, and urged reforms of tax systems to reduce the fiscal burden on labor.

Labor markets should be adapted to help workers move more easily between sectors, and the focus on the importance of education sharpened.

Dijsselbloem repeated his opposition to calls for more integration within the EU "at a time when our fundamentals are so unstable and people question the legitimacy of the EU."

At a summit on Sept 16, leaders are to consider ways of reinventing the Union after Britain voted to leave in June.

(editing by John Stonestreet)