If you're talking about herbaceous peonies (not tree peonies), my limited understanding is that Japanese peonies have single petals all around. I'm sure there's more to it than that.

Here's some info I found from Iowa State University:

"Garden peonies are classified into five different flower forms: single, semi-double, double, Japanese, and anemone.

Single flowered peonies have one row of petals surrounding a cluster of yellow stamens. The stamens are the "male" reproductive organs of the flower and are composed of the filament and pollen-bearing anthers.

Semi-doubles have multiple rows of petals surrounding a cluster of yellow stamens.

The stamens of the double-flowered peonies do not bear pollen and have been transformed into petal-like structures.

Japanese peonies have five or more large petals surrounding stamens that bear no pollen. The absence of pollen distinguishes the Japanese from the single flower form.

Anemones are similar to the Japanese type except the stamens are wider and more petal-like.

In addition to the five flower forms, the garden peony has two close relatives. They are the tree peony (Paeonia suffruticosa) and the fern leaf peony (Paeonia tenuifolia)."

Hope this clears things up!

Now I've learned something too.
Judye