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Widdershins (sometimes withershins, widershins or widderschynnes) means to take a course opposite the apparent motion of the sun viewed from the Arctic Circle, to go anti-clockwise or lefthandwise, or to circle an object by always keeping it on the left. The Oxford English Dictionary's entry cites the earliest uses of the word from 1513, where it was found in the phrase widdersyns start my hair, i.e. my hair stood on end.
The use of the word also means "in a direction opposite to the usual", and in a direction contrary to the apparent course of the sun. It is cognate with the German language widersinnig, i.e., "against" + "sense". The term "widdershins" was especially common in Lowland Scots.
Superstition and religion 
Because the sun played a highly important role in older religions, to go against it was considered very bad luck for sun-venerating traditions.
It was considered unlucky in former times in Britain to travel in an anticlockwise (anti sun wise) direction around a church, and a number of folk myths make reference to this superstition, e.g. Childe Rowland, where the protagonist and his sister are transported to Elfland after his sister runs widdershins round a church. There is also a reference to this in Dorothy Sayers's novels The Nine Tailors (chapter entitled The Second Course; "He turned to his right, knowing that it is unlucky to walk about a church widdershins, ...") and Clouds of Witness ("True, O King, and as this isn't a church, there's no harm in going round it widdershins").
In the mythology of the North Yorkshire Moors (NE England) it is believed that if you dance nine times widdershins around a fairy ring of toadstools you will come under the power of the fairy people. The story of Fairy Cross Plain (Fyup Dale) chronicles the fate of a young boy (Thomas Skelderskew) who did just that and suffered the consequences.
In contrast, in Judaism circles are sometimes walked anticlockwise. For example: when a bride circles her groom seven times before marriage, when dancing around the bimah during Simchat Torah (or when dancing in a circle at any time), or when the Torah is brought out of the Ark (Ark is approached from the right, and departed from the left).
This has its origins in the Beis Hamikdash, where in order not to get in each other's way, the Priests would walk around the Altar anticlockwise while performing their duties. When entering the Beis Hamikdash the people would enter by one gate, and leave by another. The resulting direction of motion was anticlockwise. In Judaism, starting things from the right side is considered to be important, since the right side is the side of Chessed (kindness) while the left side is the side of Gevurah (judgement). For example, it is a law to put on the right shoe first and take off the left shoe first (unless one is left-handed).
In the Eastern Orthodox Church, however, it is normal for processions around a church to go widdershins.
The Bönpo in the Northern Hemisphere traditionally circumambulate (generally) in a counter-clockwise and 'widdershins' direction, that is a direction that runs counter to the apparent movement of the Sun within the sky from the vantage of ground. This runs counter to the prevalent directionality of Buddhism (in general) and orthodox Hinduism, from which Buddhism seceded. This is in keeping with the aspect and directionality of the 'Sauvastika' (Tibetan: yung-drung), sacred to the Bönpo. In the Southern Hemisphere, the Bonpo practitioner is required to elect whether the directionality of 'counter-clockwise' (deosil in the Southern Hemisphere) or running-counter to the direction of the Sun (widdershins in the Southern Hemisphere) is the key intention of the tradition. The resolution to this conundrum is left open to the practitioner, their 'intuitive insight' (Sanskrit: prajna) and their tradition.