LONDON, July 2 (Reuters) - Euro zone bond yields were mixed on Tuesday after data showed euro zone inflation cooled in June, as the market steadied after sharp rises the previous day.

Germany's 10-year bond yield, the benchmark for the euro zone bloc, was last down 1 basis point (bp) at 2.598%, unchanged from before the inflation data. Yields move inversely to prices.

Figures showed that euro zone inflation fell to 2.5% in June from 2.6% in May, as economists expected.

Yet core inflation - which strips out volatile food and energy costs - came in at 2.9%. That was unchanged from May and above expectations for a fall to 2.8%.

"It already seemed unlikely that the European Central Bank would cut interest rates at its meeting in July, and June’s inflation data will reinforce policymakers’ inclination to move very cautiously," said Jack Allen-Reynolds, deputy chief euro zone economist at Capital Economics.

Italy's 10-year yield was up 3 bps at 4.133%, around its highest in three weeks. The gap between Italian and German bond yields rose to 153 bps.

Germany's two-year bond yield, which is more sensitive to ECB rate expectations, was down 2 bps at 2.903%.

Bond yields rose sharply on Monday as a rally in safe-haven bonds, driven by French President Emmanuel Macron's gamble in calling snap elections in early June, unwound after the results of the first round came in. Bonds are seen as more stable than stocks and preferable at times of stress.

Marine Le Pen's far right Rassemblement National party comfortably won the first round. Analysts said the market was comforted by signs that the most likely outcome is a hung parliament, however, prompting a rally in stocks and the euro.

The second round of voting is on Sunday and the final shape of the French parliament will depend on horse-trading between the groups opposing Le Pen.

France's 10-year yield was 1 bp higher at 3.355%.

The gap between France and Germany's 10-year yields - a gauge of the risk premium investors demand to hold French debt - ticked up to 76 bps. It fell on Monday as the results came in, after rising to its highest since 2012 at 85 bps last week.

The ECB's annual conference in Sintra, Portugal takes place this week, with speeches keeping investors alert.

Policymaker Pierre Wunsch said it is a relatively easy decision to cut rates again, but after that it depends on more progress on inflation. (Reporting by Harry Robertson; Editing by Andrew Heavens and Susan Fenton)