This dance was probably danced for about a hundred years before anyone bothered to publish it. The earliest Scottish source I can find is Smyth in 1830 who writes:

At the commencement, the Gentlemen place their partners at the end of the room, and stand either before or beside them.— If before them, all four begin at once, but if otherwise the Ladies must begin first, each person describing the figure eight, and the Gentlemen set to the Ladies alternately.

The Lowes in ~1831 write:

Before commencing the Reel of Four, the Gentlemen place their partners upon opposite sides of the room, or at the ends, and stand either before or beside them; if before them, all the four must begin at once; but if otherwise, the Ladies ought to begin first; each person describes the figure of eight, and the Gentlemen set to the Ladies alternately.

Arthur Miller, Leeds, 1900 writes:

In commencing the Reel of Four, the gentlemen stand back to back, with their partners opposite them, all commence at the one time. Each person describes the figure 8. The gentlemen pass the ladies on the right hand; gentlemen pass each other in the centre, pass on the left; the ladies always return to their first places, gentlemen change places every time with each other, eight bars.

All set to each other, eight bars.

Unfortunately this description is misleading. The figure of a reel for four people is not a figure eight, it's a figure eight with an extra loop. The Fletts kindly point this out.

The Fletts also mention that the reel should be performed "giving shoulders", that is when passing someone of the opposite sex in the reel the shoulder closest to that person should lead (so the dancers are almost back to back), while when passing someone of the same sex the outside shoulder should lead (so the dancers are almost face to face). The RSCDS says nothing of this, but doesn't say not to do it either...

Although the glory of the dance is in the various setting steps used, the arms were also involved. The Fletts say that the most common position was for the dancers to hold their arms akimbo. Though the gentlemen often raised their arms (one, or both, or alternating), and the ladies sometimes raised theirs, and sometimes used them to hold their skirts out.

The earliest written description of the dance actually comes from Thomas Wilson's Analysis of Country Dancing, London, 1808 (page 120). His version is rather different. It starts with two facing couples, each couple turns (I presume a two hand turn), and then everyone turns their opposite, that repeats (turn partner, turn opposite), then the 1st couple crosses in between the second and hey with them. On the second iteration the 2nd couple cross to hey with the 1st.

The RSCDS attributes this to Mozart Allan's Allan's Reference Guide to the Ball-Room, Glasgow, ~1895, and a dance with this name is in that work, but it occurred elsewhere earlier. Both Smyth and the Lowes published it in about 1830, I think Smyth was earlier.

Smyth, in 1830, page 32, writes:

Four hands across and back again, down the middle and up again, Ladies out side and back again, and pousette,

The Lowes, in 1831, page 107, writes:

Four hands across; down the middle; up again; lead out to the sides, three and three in line, (The Lady between the two Gentlemen, and the Gentleman between the two Ladies); back again, and pousette with the top couple.

These clearly describe the same dance.

It seems likely to me that the hands across, should be back again to fill out eight bars.

Both Smyth and the Lowes use "down the middle and up again" as a progressive move. (and poussette is not)

But if the Lowes are to be believed it should leave the 1s in 2nd place improper (note the 2s aren't improper, so the change can't happen in the hands across). So presumably they wheel around at the bottom rather than turning individually.

But how do the 1s become proper again? We could have the 1s do half a draw poussette, and half a regular poussette but I'm not aware of any draw poussettes in 19th century dances so that doesn't seem a good solution.

lead out to the sides ... back again usually takes 4 bars... Summing up all the figures leaves 28 bars, so perhaps a four bar movement was omitted?

Smyth (an Edinburgh dancing teacher) and the Lowes (also Edinburgh dancing teachers) classify this as a "Scotch Country Dance". Both spell the dance "Rachel Rae".

In Allan's Reference Guide to the Ball-Room, Glasgow, ~1895, the dance has become Rachael Rae.

1. The first and second couples right hands across, round and back with left hands.
2. Lead out to the sides, three and three in line, the lady between the two gentlemen, and the gentlemen between the two ladies.
3. Back again, and pousette with the top and second couples.

Courtesy of National Library of Scotland, licensed under:

The RSCDS attributes this to Allan's Ball-Room Guide, ~1880, but it appeared earlier.

Rutherford published a dance with this name in 1756, but it does not match this figure.

Smyth writes:

Four hands across half round and pousette, back to places, the first couple down the middle and turn half round and up again, four hands round at top, then turn to proper sides.

The Lowes write something similar:

Cross hands half round and pousette back to places; the first couple down the middle, and turn half round; up again and hands four at top; then turn to proper sides.

Finally Allan writes:

1. First and second couples right hand across and half round, back to places with pousette.
2. First couple down the centre, turning half round and up again.
3. First and second couples right hands across and round turning into places.

Courtesy of National Library of Scotland, licensed under:

Smyth writes:

Place the Ladies as for a set of Quadrilles, and the Gentleman in the centre. The Gentleman with the first and third Ladies make a reel of three, while the other two Ladies circle round them. All set, during which the gentleman turns to each Lady alternately, he then forms the reel of three with the other two Ladies, and set as before.

A year later, the Lowe brothers published a very similar version:

(Places — one Lady at the top of the room, one at the bottom, one at each side, and the Gentleman in the middle.) The Gentleman, with two of the Ladies, makes a reel of three, while the other two Ladies circle round them; all set, during which the Gentleman turns to each Lady alternately; he then forms the reel of three with the other two Ladies.

The earliest written description of the dance actually comes from Thomas Wilson's Analysis of Country Dancing, London, 1808 (page 125). Wilson provides two versions, the first is similar to this one, except that instead of orbiting the set during the reel, the inactive dances cross through the reel (and then cross back) instead. The other version, which he claims is the "common reel" reads:

The Lady in the middle at B, heys with the Ladies at A C, then sets to them, she then sets to the Gentlemen at F D; then heys with the Gentlemen at F D, which brings one of them into the centre, so that they all progressively occupy every situation in the figure.

Unfortunately his description of the progression omits some vital movements.

The RSCDS claims that J. Grahamsley Atkinson, Jun. was the source for this dance in his Scottish National Dances - A Practical Handbook, Edinburgh, 1900, but Smyth was there first, and Wilson before him. Atkinson doesn't mention music; the RSCDS says it should be a strathspey/reel medley.

The RSCDS attributes this dance to Mozart Allan's Allan's Reference Guide to the Ball-Room, ~1880 but it appears much earlier than that.

1. Three hands round on the ladies' side, the same on the gentlemen's side.
2. Down the centre and up again.
3. Pousette.
4. Right and left to places.

Courtesy of National Library of Scotland, licensed under:

Smyth in 1830 writes:

Three hands round on the Ladies' side, the same on the Gentlemen's, down the middle and up again, pousette, right and left once round,

In 1788 the Thompsons published an early version:

Three Ladies Promenade round the three Gent. Three Gent. round the Ladies Lead down the middle up again and cast off Allemand with your Partner