This is a description of the actual individual floral symmetry (rather than the symmetry of any flower head as a whole which 'panicle', 'globular' or 'umbel' might describe).
If the flower possesses radial symmetry, like a clock-face or magnetic compass with the line of symmetry passing vertically through the centre, then it is described as actinomorphic [radiating 'star-like' from the centre]. Examples are Daisy, Marsh Marigold or Welsh Poppy. It should be remembered that the actinomorphic symmetry ignores the asymmetry caused by Fibonacci spiral arrangement of disc florets in some flowers belonging to the Dandelion & Daisy Family (Asteraceae). Botanical symmetry is never exact anyway, petals can be blown or slightly larger than neighbours, etc.
Zygomorphic flowers possess bi-lateral symmetry, where the flower has a mirror image counterpart across a plane of symmetry, much like Himalayan Balsam or Snapdragon.
These are flowers that look like they might have radial symmetry, but actually might statistically (averaged over the population) have one (or an odd number) longer or shorter than the others, so are actually slightly zygomorphic. Examples include Brooklime one of many Speedwells which have four petals, apparently arranged radially, but one is smaller than the others and another is larger than the rest, or Mullein where one petal is slightly wider than the rest. Many individual florets on the outer extremities of the umbels of plants in the Umbellifer family are either strongly or weakly hemizygomorphic, possibly best displayed by Hogweed.
These flowers having no axis of symmetry at all. Examples are Lords-and-Ladies or Arum Lily which have twisting spathes and no petals. Strictly speaking from a botanical standpoint, these are not the flowers, but the outward appearance is one of asymmetry. They therefore possess a property called chirality, where they are either left or right handed.
Peloric flowers are genetic aberrations (or purposely bred) flowers that normally have Zygomorphic flowers but instead have (at least one) actinomorphic flowers instead. Some cultivated garden flowers are peloric. Foxglove is prone to natural pelorism.
Pelorism can be caused by one of two mechanisms. It can be developmental, in which case it will not be inheritable. Or it can be due to genetic mutation, in which case it might be an inheritable trait. The Cycloidea gene in plants controls floral symmetry. Certain Antirrhimum ( Snapdragon) cultivars are bred by horticulturalists to be peloric by knocking out this gene, because they have larger showier flowers.
Garden Gloxiniums (Sinnigia speciosa) have also been specially bred, they have large showy peloric flowers with actinomorphic symmetry whereas the original had smaller zygomorphic flowers.
FLORAL SYMMETRY and number of petals
Looking at the individual florets themselves, plants can be ordered into Monocots or Eudicots (aka Tricolpates).
Generally, Monocots have rotational symmetry of order 3, with trimeric petals and actinomorphic symmetry, possessing three or six petals, although the petals themselves may have bilateral symmetry or not. Many aquatic plants are Monocotyledons with three petals.
If the flower has 3 lines of symmetry then they belong to the Dihedral Group D3, if not, then to the Cyclic Group C3. If 4 then either to D4 or to C4; if 5 then to either D5 or C5. Regular polygons belong to the Dihedral Groups. The Three Legs of Mann symbol belongs to the cyclic C3 group because, although the legs are separated by a regular angle (120°), each leg has a foot which is directed clockwise, in a circle [it would still belong to the C3 group if the feet were instead directed anti-clockwise].
Eudicots have tetramerous (4) or pentamerous (5) petals (and their multiples, e.g. 8 or 10) and have rotational symmetry of 4 or 5, etc.