Peony bouquets: what a rushed build looks like
A peony hides poor work better than most flowers. The heads are large, they sit close together, and a full bunch reads as generous from two paces away even when the inside of it is a mess.
Look at it from the side rather than the front. The heads should form a slight dome, each one a little lower than its neighbour towards the edge, with no bloom sunk into the shoulder of another. Sunken heads mean the spiral was tied in a hurry and stems were pushed in wherever a gap showed. Within a day the buried one bruises where it presses.
Then turn the bunch upside down.
The stem ends should sit in a tight fan, cut to roughly the same length on a clean angle. Ragged ends, some crushed flat, some cut straight across, come from blunt secateurs, and a crushed stem draws water badly for the rest of its life. Peony stems are thick and full of sap, so the damage is easy to see once you know to look for it. A blackened band an inch above the cut means the stems stood in dirty water before they reached you. That one is worth raising on the day rather than after the flowers fail.
Filler, and what it is doing there
Gypsophila and eucalyptus around peonies are usually there for a reason, and the reason is not always design. In season, when stems are plentiful, a bunch of nothing but heads costs about the same as a mixed one and looks far denser. Out of season the filler is doing structural work, holding a small number of expensive blooms apart so they do not look stranded in the wrapping.
Neither approach is dishonest. It helps to know which one you are buying, because a winter bunch with a lot of greenery contains fewer peonies than the photograph suggests.
Weight, and the moment of handing it over
A peony head is heavy and the stem beneath it is soft. That combination makes a large armful awkward to hold for long, and it makes the whole bundle top-heavy, so it tips the moment it is set down on a flat surface. For a presentation on a stage or in a restaurant, where the recipient then has to shake hands and carry the thing about, a compact bunch of seven or nine heads works better than a huge one. Large armfuls belong at home, on a table, where they can be put down.
Ants come with the territory.
They feed on nectar the closed bud produces on its outer scales, they do no damage, and growers do not treat for them. A shake outdoors before the flowers come inside is enough. Their presence usually means the stems were grown in a field rather than shipped a long way, which is not a bad sign at all.
When a bouquet is the wrong format
Wrapped stems assume somebody is there to receive them, cut the ends and get them into water within the hour. If nobody is home, a box or a basket with wet foam carries its own water and buys a day or more. Offices rarely have a vase, and flowers that arrive at a reception desk often spend their first afternoon lying on their side. Long car journeys are the other weak point, since open heads bruise against each other on every corner. Closed buds travel far better, which is one quiet advantage of tight winter stems.
Peony bouquets suit a recipient in Rome who will be at home that evening and already owns a wide, heavy vase. Almost everything else about them follows from the size of the head.