I don't think we've had especially cold winters lately, but I am guessing that this hydrangea has been neglected for at least one season. The wife of the couple who sold us the house (2 months ago!) was the gardener, but she said she had been traveling a lot over the past couple of growing seasons which sort of explains why a LOT of stuff in the yard has gone to pot.

Anyway, is there anything I can do to help this hydrangea to bloom? Or should I just wait and see what happens in the spring?

Just wait to see what happens in spring/summer :-) It may have been pruned improperly, which can affect how well these bloom. And FWIW, generally in zone 7, bigleaf hydrangeas do not require any protection in winter to preserve flower buds from cold damage. Typically in these cases, the lack of bloom will be due to improper pruning, excessive shade or excessive fertilization.

Thanks! I was concerned that shade might be a factor - I know hydrangeas like some, but this one is really tucked away. I could remove that trellis behind it, but the neighbor's house still blocks the sun from the south, and the neighbor's plants have overgrown the fence farther east which blocks even more sun. Then there's a big hibiscus (I think...photo attached) on the west side that blocks the afternoon sun, so perhaps it's not an ideal location. We'll see what happens in the spring, and if I don't get blooms, I'll see if I can move the hydrangea to a more congenial setting!

Probably a mac that produces mopheads or lacecaps (lacecaps are similar to the ones at the bottom of photo #1).

If a hydrangea does not rebloom and produces blooms once a year, it would develop invisible flower buds in July-August only and these would then open in the Spring. But last winter (and the prev one too) were terrible and many of these shrubs lost their flower buds and did not bloom, which may be a reason you did not see bloomage in the summer. Almost all mac mopheads and lacecaps should be winter hardy to Zone 7 but the vagaries of winter weather may make them skip blooming in bad years. Examples of bad years would be when the temps are several USDA Zones below your normal, when temps go high (making the plant break dormancy) and then crash (killing the exposed flower buds), when there is very little rain in places were the soil does not freeze, etc.

Reblooming macs produce flower buds as described above too but, they also produce blooms again later on (say in the Summer months) but it depends on weather factors. Having the shrub produce buds in the summer means it can bloom when a bad winter kills the earlier buds.

Do not fertilize it after the end of June. Do not prune, especially after it produces flower buds somewhere in July-August. But it would be ok to prune off dead wood in mid-to-late May. Safer to prune after it blooms but before June 30th. Keep the soil as evenly moist as possible and mulch it, up to the dripline, to protect the roots and keep the soil moist longer. In the Summer, when it leafs out, you can give it 1/2 cup to 1 cup of cottonseed meal, organic compost or composted manure. Or use a chemical fertilizer with a NPK Ratio close to 10-10-10.

I'm definitely not an expert - but that second picture would appear to be a Rose of Sharon.

Indeed, I have that very same one in my back yard and the birds are spreading it to other locations... but, look on the bottom left-hand corner for some hydrangea shaped leaves.

Yes, hydrangea and Rose of Sharon is the consensus. Thanks everyone!