Square dance formation terminology

For the purpose of attempting to illustrate the various formations used in modern western square dancing, the following symbols and abbreviations will indicate dancers facing in the direction of the symbol's open side (imagine that the arms of the symbol are the dancer's outstretched arms), i.e.,

⊃

dancer facing left, relative to the caller

⊂

dancer facing right, relative to the caller

∪

dancer facing away from the caller

∩

dancer facing the caller

RH

right-handed formation

LH

left-handed formation

Note: If you are viewing this on a computer screen, you are probably looking at it in a vertical plane; real dancers, obviously, would be oriented in a horizontal plane on the dance floor. Also, all that's really critical to a proper formation is how the dancers are oriented relative to one another. Formations rotated 90°, 180°, and 270° from these depicted are still the same formation, even though different dancers or groups of dancers would then be facing toward or away from the caller, or to his/her right or left.

Standard formations generally are symmetrical, as opposite counterparts in the square are usually doing the same sequence of movements, which creates a mirroring effect. Asymmetrical formations are only possible if the caller specifically directs opposite counterparts in the square to do two different calls at the same time (e.g., normally a caller might call something like "head couples, lead to the left and circle up four"; both head couples would do the same movements, creating a symmetrical arrangement. If, however, the caller called something like "couple 1 lead to the left and circle up four; while couple 3 lead to the left and square chain thru," the resulting formation would no longer be symmetrical. Asymmetrical sequences are usually only attempted at higher square dance levels with experienced dancers.)

Couple
∪∪ - therefore, also ∩∩, etc.

Tandem
∪
∪

Facing dancers
⊂⊃ - also ∩ , etc. 
           ∪

Back to back dancers
⊃⊂

Mini-wave (RH)
∪∩

Mini-wave (LF)
∩∪

Facing couples
∩∩
∪∪

Back to back couples
∪∪
∩∩

T-Bone box (typical)
∩∩
⊂⊃

Inverted box
∩∪
∪∩

Ocean wave (RH)
∪∩∪∩

Ocean wave (LH)
∩∪∩∪

Box circulate (RH)
∪∩
∪∩

Box circulate (LH)
∩∪
∩∪

Two-faced line (RH)
∪∪∩∩

Two-faced line (LH)
∩∩∪∪

Static (starting) square
  ∩∩
⊂    ⊃
⊂    ⊃
  ∪∪

Circle
(like the static square, but with everyone offset slightly to make a circle,
and facing toward the center -- I know no way using HTML characters to indicate 
the acute and obtuse angles necessary to show actual facing for this and 
related formations)

Single file promenade 
  ⊃⊃
∩    ∪
∩    ∪
  ⊂⊂

Alamo style
  ∩∪
⊃    ⊃
⊂    ⊂
  ∩∪

Promenade
   ⊃
   ⊃
∩∩  ∪∪
   ⊂
   ⊂

Wrong-way promenade
   ⊂
   ⊂
∪∪  ∩∩
   ⊃
   ⊃

Star (RH)
  ⊂
∪  ∩
  ⊃

Star (LH)
  ⊃
∩  ∪
  ⊂

Thar star
   ⊃
   ⊂
∩∪  ∩∪
   ⊃
   ⊂

Wrong-way thar
   ⊂
   ⊃
∪∩  ∪∩
   ⊂
   ⊃

Facing lines
∩∩∩∩
∪∪∪∪

Eight chain thru
⊂⊃⊂⊃
⊂⊃⊂⊃

Trade by
⊃⊂⊃⊂
⊃⊂⊃⊂

Double pass thru
⊂⊂⊃⊃
⊂⊂⊃⊃

Completed double pass thru
⊃⊃⊂⊂
⊃⊃⊂⊂

Lines facing out
∪∪∪∪
∩∩∩∩

Parallel waves
∪∩∪∩
∪∩∪∩

Parallel two-faced lines
∪∪∩∩
∪∪∩∩

Columns (RH)
⊂⊂⊂⊂
⊃⊃⊃⊃

Columns (LH)
⊃⊃⊃⊃
⊂⊂⊂⊂

Three and one lines (8 variations possible)
∪∪∪∩
∪∩∩∩

Inverted lines (ends in)
∩∪∪∩
∪∩∩∪

Inverted lines (ends out)
∪∩∩∪
∩∪∪∩

T-Bone lines (in)
⊂∩∩⊃
⊂∪∪⊃

T-Bone lines (out)
⊃∩∩⊂
⊃∪∪⊂

1/4 tag
  ⊂
⊂⊃⊃
⊂⊂⊃
  ⊃

3/4 tag
  ⊂
⊃⊃⊂
⊃⊂⊂
  ⊃

1/4 line
  ⊂
⊂⊂⊃
⊂⊃⊃
  ⊃

Diamonds (RH)
 ⊂    ⊂
∪ ∩ ∪ ∩
 ⊃    ⊃

Diamonds (LH)
 ⊃    ⊃
∩ ∪ ∩ ∪
 ⊂    ⊂

Point to point diamonds (RH)
   ⊂       ⊂
∪    ∩ ∪    ∩
   ⊃       ⊃

Facing diamonds (RH)
 ⊂    ⊂
∩ ∪ ∩ ∪
 ⊃    ⊃

Facing diamonds(LH)
 ⊃    ⊃
∪ ∩ ∪ ∩
 ⊂    ⊂

Hourglass
 ⊂ ⊂
  ⊂
∪   ∩
  ⊃
 ⊃ ⊃

Galaxy
  ⊂
 ⊂ ⊂
∪    ∩
 ⊃ ⊃
  ⊃

Butterfly (typical)
∩  ∩
 ∪∪
 ∩∩
∪  ∪

O (typical)
 ∩∩
∪   ∪
∩   ∩
 ∪∪

H (typical)
∪    ∩
∪∪∩∩
∪    ∩

Blocks (typical)
  ∩   ∩
∩   ∩
  ∪   ∪
∪   ∪

Phantom (typical)
 ⊂
     ∩∩
 ⊂
      ⊃
∪∪
      ⊃

Siamese (typical)
 ∩
     ⊂⊂
 ∩
      ∪
⊃⊃
      ∪

Offset blocks (typical)
  ⊃⊃
  ⊃⊃
⊂⊂
⊂⊂

Tidal wave (RH)
∪∩∪∩∪∩∪∩

Tidal two-faced line (RH)
∪∪∩∩∪∪∩∩

Tidal line
∪∪∪∪∩∩∩∩

Z (typical - RH)
∪
∪∩
  ∩

Z (typical - LH)
∩
∩∪
  ∪

Source: my own experience as a dancer, with my memory jogged by the Callerlab formation listings on www.penrod-sq-dancing.com