There are orange deciduous azaleas available in the trade - that is what you are looking at here.

You might check here:
http://www.rarefindnursery.com/index.cfm/action/displayProducts/level/2:4.htm

And here: http://www.rhodyman.net/rhodyde.html

Depending on where you live and the lowest temperatures for your area, the choice may vary. As you can see in the second link, there are Ghent hybrids, Exbury/Knap Hill hybrids and Northern Lights. Where I live (Georgia) there are actually species azaleas that would work (R. flammeum, R. calendulaceum) for a similar look.

Since we don't know where the plant are located and when they bloom, it is very difficult to guess what they are. Here are some possibilities

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Austrinum blooms in early spring and is found in the wild north of the gulf coast. It is fragrant.

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Calendulaceum comes is a wide variety of colors from yellow through orange to red. It blooms in late spring or early summer. It is found over a wide area in the Appalachian mountains. It has a clove scented fragrance.

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Cumberlandense is found over a wide area of the Appalachian mountains also. It is very similar to calendulaceum but it blooms after the leaves are fully expanded. It tends to have larger flowers.

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Flammeum is found in the southern Appalachians. It blooms soon after austrinum but the flowers have a blotch and are not fragrant. It is probably not this.

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Klondyke is a popular hybrid Exbury azalea. It typically blooms in early summer. The Exbury azaleas are among the oldest hybrids and were created in a breeding program that started in 1922. That means that yours is most likely not a hybrid but one of the species above. The time of bloom in your location and any fragrance would be key in identifying it.

Here is a link that might be useful: Deciduous Azaleas

Thank you for the suggestions!
It is located in Southeast Michigan, and the photos in bloom were taken on Mother's day this year. It had a very noticeable sweet fragrance.

Then it sounds a lot like 'austrinum' but since austrinum is considered hardy to -5F, it is probably not this species growing in SE Michigan, but rather a hybrid. The closest hybrid I know is the Exbury hybrid 'Klondyke' from the Exbury Estate in Southhampton, England, which was not introduced until 1947. Klondyke is considered hardy to -15F. Just as hardy, but before the Exbury deciduous azaleas were the Knapp Hill deciduous azaleas hybridized by Anthony Waterer at Knapp Hill, England. One Knapp Hill, that looks similar to this, was obtained by Lionel de Rothschild and introduced in 1936. It is 'George Reynolds'. So if it truly is from the '30s, it could be 'George Reynolds.' But if it could be from after 1947, it could also be 'Klondyke.' It could also be an unnamed seedling. Both Exbury and Knapp Hill sent out seed from their plants. In any case, it is correct to call it a "Knapp Hill Azalea" since Exbury azaleas are technically Knapp Hill azaleas.

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George Reynolds (photo from Harold Greer at Greer Gardens in Eugene, OR.)

Very informative! Since many parts of the estate were brought over from England (stair cases, wall paneling, stone, roofing tiles, etc.) it would not be the least bit surprising if some of the landscaping was also from England.