I learned about sitting square dances from Rick Hill at Ashokan Fiddle and Dance Camp, Southern week. He learned about them from Glenn Bannerman who taught recreation at the Presbyterian School of Christian Education.
Here are the figures that I learned from Rick and have made up on my own so far. They are written in the context of squares, but most would work the same in a circle or even sitting in a line. Some clearly require partners and some assignment of head/side or active/inactive, but others would work fine without partners. Many dances now use gender neutral terms instead of "gents" and "ladies". Do what works for you.
Circle: Hold hands in a circle. Circle arms in a jump-rope type of motion. Lean left for circle left, lean right..., lean forward for into the middle, then lean back.
Forward and back: individual or couple lean forward, then back to upright, locked position.
Do-si-do: Fold arms in 3rd - grade style do-si-do. Facing more or less toward corner or partner, lean forward and to left, then arc around to right and back to upright.
alternate dosido - for people who like to twirl - Bend elbows so forearm is pointing up and circle fingers while doing the same lean-arc-back motion.
Comment: can also do-si-do to other side (lean right first)
Pass Through: With arms at sides, bend elbow at 90 degree angle so that forearms are facing forward. Scissors arms back and forth.
Swing: Hold both hands of partner and swing arms sideways back and forth.
Promenade: Hold partner in promenade position. Gent's right hand on ladyís right shoulder is probably the best - or any comfortable method. Couple rocks to left and right.
Allemande l and r: just shake left or right hands with corner/partner.
Grande allemande: Gents right allemande corner and left allemande partner at same time (could work other way, but gents on average may have longer arms)
Cross trail: Lady leans forward and to left and straightens, then gent leans forward and to right and straightens
Courtesy turn: Gent raises partners handy hand. Lady circles index finger under gentís raised hand. (This almost never works, so not using it anymore.)
Hot Potato: Everyone holds left hand out to side at about chest height with palm up. Then everyone holds right hand to other side with palm up and rests their right hand in the left palm of their partner or corner. Call is to head ladies or gents or side ladies or gents or all four ladies or gents (but not to everyone at the same time) Head ladies take right hand and gently slap the right hand that is resting in their left hand, then return their right hand to the original location. The person whose hand was slapped then slaps the hand in his left hand and returns. And so on around the circle.
Grand right and left: Shake right hands with partner, left with corner, right with partner and so on.
Weave the ring: Face corner. Stretch arms in front, palm to palm as if you were diving. Lean and stretch to left. Then lean and stretch to right, and so on. Gent and corner leaning in opposite directions so that they don't hit each other.
Cross stitch: Everyone claps hands on knees twice (right hand on right knee, left on left). Then everyone claps hands together twice. Cross arms and clap opposite knee twice. Clap hands with persons to either side twice (if gent, for example, right hand to partner, left hand to corner). A nice 8-beat figure.
Balance: Hold handy hand and lean toward partner and back.
Hey: Heads hold hands with each other and corner (Form lines at the heads) Everyone yells Hey Hey Hey Hey. Can be done by forming lines at sides too, of course.
Roll in the Hey: Hold partner by handy hand. Gent or lady leans away then rolls back to place. Everyone calls Hey Hey Hey Hey while doing this.
Dip and Dive: Sides arch. Heads hold hands and lean forward. Then switch.
Wave: Like in a baseball stadium. One person can start, or a couple, or heads or whatever. Go around the ring. Not a traditional figure - but so what.
Star: Left or right arm out in front at shoulder height. Make small circles with arm.
Ladies chain: Half right hand star followed by courtesy turn. (This never worked well because of the faulty courtesy turn, so also avoiding.)
Four ladies chain: All ladies right hand star half way and courtesy turn.
Mirror (Formerly called Gypsy, but that name has fallen out of use in my area.): Look at partner. Both move heads (and shoulders) in a circle while maintaining eye contact. Gents clockwise, ladies counterclockwise. (This is a mirroring move between partners.)
Whirlpool: Star left and right at the same time.
Milk the cow: Both arms straight out in front and scissor up and down several inches.
California twirl: Turn to partner and in best imitation of The Terminator, say: "I'll be back." Then turn to corner. Think about switching partners and maybe not going back. But then realize that you've got to honor your commitment and turn back to your partner. Raising and resolving the moral dilemma is the key to making this a meaningful figure.Some example dances (a PDF file.) Note: In the tables, the left column is the main figure. The right is the break. The intro is at the bottom. This is because I haven't memorized all the calls.