Home prices are on the rise again, a key index released Tuesday morning confirmed, with housing values up 0.44 percent in 20 major cities in July, as measured by the Standard & Poor’s/Case-Shiller index.

But that report, which also showed a 1.2 percent gain over the past year, masks a housing market that is flashing wildly different signs in different places in the United States, from down 9.9 percent in Atlanta to up 16.6 percent in Phoenix.

The divergent trends in home prices in those two cities — and the 18 that fall somewhere between — are tied to the peculiarities of each metro area and show why for all the government efforts to get the housing market back on track, such as mortgage relief programs and low interest rates out of the Federal Reserve, any bounce back in the sector has been halting and uneven.

Housing is funny. In theory, it is a small piece of the economy. Residential investment has averaged less than 5 percent of economic activity over the past 60 years and is currently a minuscule 2.2 percent of gross domestic product.

Yet it has a strangely dominant role in fueling, or even creating, economic swings and has an even larger place in the American psyche. It is the most volatile portion of GDP, almost always swinging wildly and making economic highs higher and lows lower. It is a consumer good and a financial asset: People buy homes to live in but also expect them to be a store of value over time. The mortgage loans for which homes serve as collateral form part of the backbone of the financial system, as the world learned all too well starting in 2007.

Home prices show continued improvement. (The Washington Post/Source: S&P/Case Shiller)

Local governments place zoning and other land-use restrictions on how many homes can be built, which means that supply will not necessarily keep up with demand. (If demand for Ford Focuses rises, Ford builds more of them; if demand for condominiums in San Francisco rises, if more are built at all, they will come only after long and expensive delays.) Finally, home sales have an outsize effect on other consumer spending; someone who buys a new house is likely to fill it with furniture and appliances, for example.

Given all that, the old saw that all politics is local could be applied even more clearly to housing.

For the 20 metro areas covered by the Case-Shiller index, I pulled data on job growth over the past year and housing permits issued to get a sense of how these three measures — one of home prices, one of job growth, another of change in the rate of housing construction — interrelate.

The simple answer is, unpredictably. You might expect, for example, a tight relationship between job growth and changes in home prices. In fact, the correlation between those numbers is only about 40 percent. (The relationship between change in housing starts and change in home prices is even weaker, with a 21 percent correlation.)

But that doesn’t mean there are no patterns.

Solid economies, housing weakness

First, the rate of home construction has surprisingly little to do with prices. The worst market, Atlanta, featured a 9.9 percent drop in home prices over the past year, yet the number of housing permits issued in the first seven months of 2012 was up a whopping 78 percent from the same period a year earlier. Atlanta appears to be a case in which builders are getting ahead of the market; it is a city with middling job growth, yet they are building homes at a breakneck pace, sending prices down.

Interestingly, Washington area homeowners may want to watch out, lest they become the next Atlanta. The Washington metropolitan area, after years as the star among local economies, has slowed down in its pace of job creation. With payrolls up only 0.8 percent in the year ended in July — fourth worst among the 20 major metros — the 3.7 percent gain in home prices in Tuesday’s report may not be sustainable given steady construction levels in recent years. Some other major markets that have had decent economic performance in the past few years may already be there. The New York and Chicago metro areas had job growth rates about on par with Washington yet have seen home price declines over the past year.

Bubble markets coming back

Among the Sun Belt cities that were nexuses of the housing bust a few years ago, the results today are highly sensitive to whether jobs have come back. Las Vegas, with only 0.9 percent job growth, is the second worst market for home price declines. Meanwhile, Phoenix, which experienced a bubble roughly as large but is now seeing job growth of 2.9 percent, is the best city for home price rises; the typical home has seen its price rise 11.5 percent in the past year, though from a foreclosure-depressed low. After years of very little home construction activity, Phoenix is even starting to see a bit of a building boom, with permits up 70 percent over the first seven months of the year.

Other onetime bubble markets seem to be following the Phoenix pattern, if less dramatically: Tampa and Miami are seeing solid home price gains of 3.6 percent and 5.3 percent, respectively, on the back of modest job growth.

Dreary economies, mixed recovery

The story is more dreary in the depressed cities of the industrial Midwest, where improvement in the auto and other manufacturing sectors has not translated into higher prices. Detroit has managed decent job growth after decades of decline, with a 1.8 percent gain that has translated into a strong 6.2 percent rise in home prices — though that is off of a depressed base. And there’s less momentum in housing construction. Although a 35 percent gain might sound impressive, it translates to only 2,492 housing permits in the first seven months of the year in the metro area of 4.2 million people.

Cleveland is worse, with weaker job growth, a mere 0.4 percent rise in home prices, and pitiful levels of home construction. (About 1,175 permits were issued in the metro area of 2.9 million.)

The superstars

Then there are the stars of the housing recovery. These are cities that are recording decent growth in jobs, housing starts and home prices. Put those together, and it points toward a healthy environment: home price increases driven by economic fundamentals, not a bubble mentality, and in places where builders are being allowed to construct homes to answer that demand.

The big winners, for both their local housing markets and the U.S. economy as a whole, are Minneapolis, Dallas, Denver, Charlotte, San Francisco, Seattle and Portland, Ore. If the rest of the country looked like them, pretty soon, it wouldn’t feel like a recession any more.