Floral symmetry

Floral symmetry describes whether, and how, a flower, in particular its perianth, can be divided into two or more identical or mirror-image parts.
Uncommonly, flowers may have no axis of symmetry at all, typically because their parts are spirally arranged.
Actinomorphic.
Most flowers are actinomorphic ("star shaped", "radial"), meaning they can be divided into 3 or more identical sectors which are related to each other by rotation about the centre of the flower. Typically, each sector might contain one tepal or one petal and one sepal and so on. It may or may not be possible to divide the flower into symmetrical halves by the same number of longitudinal planes passing through the axis: Oleander is an example of a flower without such mirror planes. Actinomorphic flowers are also called radially symmetrical or regular flowers. Other examples of actinomorphic flowers are the lily ("Lilium", Liliaceae) and the buttercup ("Ranunculus", Ranunculaceae).
Zygomorphic.
Zygomorphic ("yoke shaped", "bilateral" – from the Greek ζυγόν, "zygon", yoke, and μορφή, "morphe", shape) flowers can be divided by only a single plane into two mirror-image halves, much like a yoke or a person's face. Examples are orchids and the flowers of most members of the Lamiales (e.g., Scrophulariaceae and Gesneriaceae). Some authors prefer the term monosymmetry or bilateral symmetry. The asymmetry allows pollen to be deposited in specific locations on pollinating insects and this specificity can result in evolution of new species.
Asymmetry.
A few plant species have flowers lacking any symmetry, and therefore having a "handedness". Examples: "Valeriana officinalis" and "Canna indica".
Differences.
Actinomorphic flowers are a basal angiosperm character; zygomorphic flowers are a derived character that has evolved many times.
Some familiar and seemingly actinomorphic so-called flowers, such as those of daisies and dandelions (Asteraceae), and most species of "Protea", are actually clusters of tiny (not necessarily actinomorphic) flowers arranged into a roughly radially symmetric inflorescence of the form known as a head, capitulum, or pseudanthium.
Peloria.
Peloria or a peloric flower is the aberration in which a plant that normally produces zygomorphic flowers produces actinomorphic flowers instead. This aberration can be developmental, or it can have a genetic basis: the CYCLOIDEA gene controls floral symmetry. Peloric "Antirrhinum" plants have been produced by knocking out this gene. Many modern cultivars of "Sinningia speciosa" ("gloxinia") have been bred to have peloric flowers as they are larger and showier than the normally zygomorphic flowers of this species.
Charles Darwin explored peloria in "Antirrhinum" (snapdragon) while researching the inheritance of floral characteristics for his "The Variation of Animals and Plants Under Domestication". Later research, using "Digitalis purpurea", showed that his results were largely in line with Mendelian theory.
Symmetry groups.
If we consider only those flowers which consist of a single flower, rather than a flower head or other form of inflorescence, we can categorize their symmetries into a relatively small number of two-dimensional symmetry groups. These groups are characterized by two types of symmetries: reflection (or mirror) symmetries, and rotational symmetries. Figures that are left invariant under reflections about a single axis have reflection symmetry, which is described by the cyclic group of order 2, formula_1 (sometimes denoted formula_2). Figures that are left invariant under rotations by formula_3 have a rotational symmetry belonging to the cyclic group of order formula_4, formula_5 (or formula_6). Many flowers that are invariant under rotations by formula_3 are also invariant under reflections about formula_4 distinct axes, the combination of these two symmetries forms the larger dihedral group of dimension formula_4, formula_10 (which has order formula_11).
Flowers with bilateral symmetry, such as orchids have reflection symmetry about a single axis and no rotational symmetry, meaning that they are described simply by the reflection group formula_1.
Monocots are identifiable by their trimerous petals, meaning that they are often invariant under rotations by formula_13 and thus have rotational symmetry. Monocots that exhibit rotational symmetry but not mirror symmetry (for instance, if their petals exhibit chirality) are described by the cyclic group of order 3, formula_14, and monocots with both rotational symmetry and reflection symmetry about 3 axes are described by the dihedral group of dimension 3, formula_15.
Eudicots with tetramerous or pentamerous petals are often invariant under rotations by formula_16 or formula_17. Again, whether they also have mirror planes decides whether they belong to dihedral (formula_18 and formula_19) or cyclic groups (formula_20 or formula_21). 
We can see the trend forming that, in general, the order of the cyclic group or dimension of the dihedral group that describes a flower's symmetry will correspond to the merosity of its petals. However, the sepals of some monocot flowers develop to replicate the petals, thus, superficially, certain monocots can appear to have rotational symmetry of order 6 and belong to either symmetry group formula_22 or formula_23. Some composite flowers may also have at least a superficial cyclical or dihedral symmetry. How exact this symmetry is depends on the structure of the head of the flower. Even in monocots and eudicots, flower symmetries are rarely perfect, as any imperfections in the petals will result in imperfect invariance under rotations or reflections.