These directions may be taken either facing or backing. For example, "backing DC against LOD".
These are described in one of the following ways.
The direction of a turn is indicated in one of the following ways.
While the first two ways are fairly straightforward, the third one requires some explanation, and the fourth one is often a source of confusion.
A "natural" theory of the origin of the term is that considering the right-shifted position in a couple (even more pronounced in older times) and the counterclockwise direction of travel along the line of dance, the right turns are easier to perform, they are more "natural". This is especially true at the corners of the LOD: the amount of the right turn is effectively 3/4 of a full turn, and the amount of the left turn is effectively 5/4, because the LOD changes its direction by 90 degrees to the left (CCW).
The term is used in the names of figures and is mostly applied to an individual turn of a partner in the couple. Basically, it denotes the turn directed "inside" or "outside" of the couple. The meaning is intuitively clear, but it may be performed in numerous ways and in different handholds, so that even accomplished dancers are confused as to the name.
Most often it is understood that an inside turn is an underarm turn under the right arm if turning left and under the left arm if turning right. Respectively, the outside turn is in the opposite way. This is intuitive clear if initially the couple is in an open single-handhold position facing each other, and the name corresponds to the direction of the lead. To lead the inside turn, the leader moves the follower's arm "inside". The similar rule holds for the "outside". When the position is significantly different, the term is not so clear.
Therefore in some dances some prefer to use this term according to its usage in ballet, based on the footwork, rather than arm style. In ballet, when describing pirouettes, an outward (en dehors) turn is the turn in the direction towards the working leg. Accordingly, an inward (en dedans) turn is the turn in the direction towards the support leg. See also Ballet glossary#Ronds de jambe.
Although the latter definition has the benefit of unambiguity, its drawback is that in other contexts it is applicable to a single footstep only. For example, according to this definition, the Chaines turns is an alternating chain of inside of outside turns, although the direction of the rotation is the same. Therefore it is common to name the turning figure according to the direction of the first turning step.
Concluding, the terms "inside/outside turn" should be read only in the exact context of the dance, and possibly, the particular dance school.