The Trinacria
Trinacria is both an alternative name for Sicily and its national symbol (an ancient form of the Triskelion) which also appears on its flag.
Historically, the name was used after the Peace of Caltabellotta (1302) to distinguish between the two Sicilies that became a reality after the War
of the Sicilian Vespers.
Frederick III of Sicily was recognised as king over Sicily, with the title of "King of Trinacria".
Charles of Anjou was recognised as king over the southern portion of the Italian peninsula, with the title of King of Sicily, usually rendered by historians as "King of Naples," for such was his capital.

  The Mythology
  The Medusa in the center implies the protection of the Goddess Athena, the Patron Goddess of the Isle.
  In Greek mythology, Medusa (Greek: Μέδουσα, "guardian, protectress") was a monstrous chthonic female character; gazing upon her could turn
  onlookers to stone.
  She was beheaded by the human hero Perseus, who thereafter used her head as a weapon until giving it to the goddess Athena to place on her
  shield.
  In classical antiquity and today, the image of the head of Medusa finds expression in the apotrope known as the Gorgoneion.

The Triskelion
A triskelion or triskele (Greek for "three-legged") is a symbol consisting of three interlocked spirals, or three bent human legs, or any similar symbol with three protrusions and a threefold rotational symmetry.
A triskelion is the symbol of Brittany, as well as the Isle of Man and Sicily (where it is called Trisceli).
The Manx and Sicilian triskelions feature three running legs, bent at the knee and conjoined at the crotch.
Spiral forms of the triskele are often classed as solar symbols, while the legged version, sometimes including a gorgon mask or Medusa's head at the central axle point in the Sicilian version, suggests a chthonic significance.

  The Origins
  The triskelion symbol appears in many early cultures, including on Mycenaean vessels, on coinage in Lycia, and on staters of Pamphylia
  (at Aspendos, 370-333 BC) and Pisidia.
  A symbol of four conjoined legs, a tetraskelion, is also known in Anatolia.
  Celtic influences in Anatolia, epitomized by the Gauls who invaded and settled Galatia, are especially noted by those who theorize a Celtic origin for the
  triskelion.

The Manx Triskelion
In the Isle of Man' symbol, which is located in the Irish Sea, the "three legs embowed" of the heraldic triskelion are represented in armour, "spurred and garnished.
"On Manx banknotes, the triskelion appears within a rim containing the Latin inscription QUOCUNQUE JECERIS STABIT ("Wherever you throw it, it stands").
The Manx triskelion is documented since the 13th or 14th century at the latest, and it is alternatively known in the Manx language as the tre cassyn
("three legs").
The symbol appears on the Isle of Man's ancient Sword of State, which may have belonged to Olaf Godredson, who became King of the Sudreys (Southern Hebrides and the Isle of Man) in 1226. Kneeling clockwise is the correct symbol as it is said that bent or kneeling in an anti-clockwise fashion signifies aggression in heraldry. However, many Manx Churches show the anti-clockwise version.

  The Sicilian Triskelion
  Familiar as an ancient symbol of Sicily, the triskelion is also featured on Greek coins of Syracuse, such as coins of Agathocles (317-289 BCE).
  In Sicily, the first inhabitants mentioned in history are the tribes of the Sicani and the Siculi, who have given Sicily its more familiar modern name.
  The triskelion was revived, as a neoclassic-and non-Bourbon-emblem for the new Napoleonic Kingdom of the Two Sicilies, by Joachim Murat in
  1808. The symbol dates back to when Sicily was part of Magna Grecia, the colonial extension of Greece beyond the Aegean. Pliny the Elder
  attributes the origin of the triskelion of Sicily to the triangular form of the island, the ancient Trinacria, which consists of three large capes
  equidistant from each other, pointing in their respective directions, the names of which were Pelorus, Pachynus, and Lilybæum.
  The three legs of the triskelion are also reminiscent of Hephaestus's three-legged tables that ran by themselves, as mentioned in Iliad xviii: "At the
  moment Hephaestus was busily turning from bellows to bellows, sweating with toil as he laboured to finish a score of three-legged tables to stand
  around the sides of his firm-founded hall.On each of the legs he had put a gold wheel, that those magic tables might cause all to marvel by going
  with no other help to the gathering of gods and by likewise returning to his house."

The Spiral Triskele
The Celtic symbol of three conjoined spirals may have had triple significance similar to the imagery that lies behind the triskelion.
The triple spiral motif is a Neolithic symbol in Western Europe. It is carved into the rock of a stone lozenge near the main entrance of the prehistoric
Newgrange monument in County Meath, Ireland.
A variant of the symbol is also found, carved into the wall in the inner chamber of the passage tomb. Because of its Celtic associations, it is also used as a symbol of Brittany (alongside the hermine).
In the north of Spain, the triskelion is used as a symbol of Galizan and Asturian nationalists.
A similar symbol called lábaro by Cantabrian regionalist can be compared to the neighboring Basque culture's four-branched lauburu.
A possibly related symbol of Germanic origin is the valknut, and the Celtic and Germanic triquetra. 

  The Third Reich
  Das Dritte Reich ("The Third Reich") is a 1923 book by German author Arthur Moeller van den Bruck, the ideology of which heavily informed the Nazi party.
  The book formulated an "ideal" of national empowerment, which resounded throughout a Germany desperate to rebound from the Treaty of Versailles: Das Dritte Reich
  was Germany's Third Rome.
  For Moeller van den Bruck, Germany's great misfortune lies in the political system created by the Weimar Republic, one of competitive parties and liberal ideologies.
  An admirer of Mussolini, he looks for a strong leader. The Third Reich adopted a variation on the triskelion as the insignia for a Waffen SS division composed of Belgian
  volunteers. It is claimed, possibly apocryphally, that the similarity to the swastika caused confusion or distress amongst some Jewish refugees interned on the Isle of
  Man during World War II.

Reconstructionists and Neopagans
The triskele, usually consisting of spirals, but also the "horned triskelion", is used by some Polytheistic Reconstructionist and Neopagan groups.
As a Celtic symbol, it is found primarily of groups with a Celtic cultural orientation and, less frequently, can also be found in use by some Germanic Neopagan groups and eclectic or syncretic traditions such as Wicca.
The spiral triskele is one of the primary symbols of Celtic Reconstructionist Paganism. Celtic Reconstructionists use the symbol to represent a variety of triplicities in their cosmology and theology; it is also a favored symbol due to its association with the god Manannán Mac Lir.
Wicca is syncretic in nature and often aesthetically adopts symbolism from various cultures, particularly Celtic symbolism.
It is less commonly used amongst Germanic Neopagan groups due to the non-Germanic origins of the symbol; use by Germanic groups may be due to confusion or association with Norse symbols with triple symmetries, like the Valknut, the Triquetra, or the symbol found on the Snoldelev Stone. 

  BDSM
  The term "BDSM" is an abbreviation derived from the terms bondage and discipline, domination and submission and sadism and masochism.
  It defines a spectrum of usually sexual behavior, that can include dominance, submission, punishment, masochism, bondage, role play and a large variety of
  other activities.
  BDSM can also be referred to as "kinky sex", "(consent) power exchange", "fetish", "SexMagick" or "the lifestyle".
  A form of the triskelion has been proposed as a BDSM Emblem by some BDSM groups, partly based on a description in the Story of O.
  The specific emblem design is meant to be shown with metallic spokes and circle, and three holes (not dots) within the design. 

Political Groups
The South African white supremacist paramilitary group, the Afrikaner Weerstandsbeweging (AWB), have used a flag consisting of a red background with a white circle.
In the circle, three black sevens form a design distantly reminiscent of the triskelion.
In spite of the similarities to the swastika (and the overall resemblance to the Nazi flag, both having a black symbol on a white disc on a red background), they claim their flag is inspired by a Biblical meaning of the seven, and the fact that the organisation was founded on the seventh day of the seventh month, 1973 (from which the three 7's can be extracted). 

  Other Uses
  A triskelion shape is the basis for the Roundel of the Irish Air Corps (unique among air force roundels). It is loosely based on the Flag of Ireland and traditional
  Celtic triskele boss designs.
  The Triskelion is shown on the seal of Tau Gamma Phi, the most dominant fraternity in the Philippines. A member of Tau Gamma Phi is called a Triskelion.
  The triskelion is accompanied by the motto "FORTIS VOLUNTAS FRATERNITAS" on the seal.
  A triskelion pattern forms part of the seal of the United States Department of Transportation. The three spirals represent air, land, and sea transportation.
  The seal was adopted on February 1, 1967.
  Triskelion is the name of the GSA at Brandeis University, one of the oldest such organizations on the East Coast. It was named for the Star Trek episode.
  A fractal version of the triskelion, consisting of a large blue-silver raised dot with three curved arms of similar dots around it, is a major motif of the 2005 TV
  series Threshold. It is the symbol of the aliens who invade Earth.