WASHINGTON — Home prices in 20 cities climbed in June for the first time since a tax credit boosted sales in 2010, indicating the industry at the heart of the worst recession in the post-World War II era is starting to rebound.

The S&P/Case-Shiller index increased 0.5 percent from June 2011 after falling 0.7 percent in the year to May, a report from the group showed Tuesday in New York. The last 12-month increase took place in September 2010. Nationally, prices jumped last quarter by the most in more than six years.

The lowest mortgage rates on record and a decline in sales of distressed properties may help the market contribute to the economic expansion that is now in its fourth year. A more sustained rebound may require easier lending conditions, which would also give consumers a lift after a report today showed household confidence sank to the lowest level of the year.

“Finally, the housing market is forming a bottom,” Mohamed El-Erian, chief executive officer and co-chief investment officer of Pacific Investment Management Co., said on Bloomberg Television’s “In the Loop” with Betty Liu. “That should be welcome. It is not surprising because affordability is so attractive right now.”

Overseas, housing markets aren’t faring as well. Sales of newly built homes in Australia dropped in July to the second- lowest level on record, a report Tuesday showed.

The S&P/Case-Shiller 20-City index was projected to drop 0.05 percent in the year to June, according to the median forecast of 29 economists surveyed by Bloomberg. Estimates ranged from declines of 1.5 percent to a 1 percent gain. The index is based on a three-month average, which means the June data were influenced by transactions in April and May.