CALLER HERRING
https://www.regencydances.org/index.php?wL=2530

Caller Herring (Button & Whitaker)
 Longways Country Dance, duple minor
Button & Whitaker's Selection of Dances Reels and Waltzes No 11, c.1809

Music (treble clef, bass clef, no chords) at 
https://archive.org/details/ButtonWhitakerDancesReelsAndWaltzesNo11/page/n1/mode/2up

  
 
8 bars	The first couple join hands with the second lady and circle once around, 
        then with the second man
4 bars	The first couple join hands and slip down the middle and 
        return to second place, seconds moving up
4 bars	Swap diagonals at the top passing near shoulder each time; 
        begin with the first man and second lady, the couples passing back-to-back


Caller Herring (Wheatstone)
 Longways Country Dance, duple minor
A Selection of Elegant & Fashionable Country Dances, Reels, Waltz's &c. Book 3
  
DANGER, DANGER, 28 bar sequence amd the notation must be differenct.
 
8 bars	The first and second couples set to their neighbour, 
        the ladies join hands and change sides with the men; 
        then set to their neighbour and return

8 bars	The first couple lead down the middle and back, and cast off into second place

12 bars	The top two couples join hands and circle once around, and back again

[Regencydances.org circles all the way in 8 counts, turns single in 4, return all the way in 8, turn single for 4.]
This implies that you could use the Button &Whitaker movement and just have a regular circle left / circle right.