The Flirtation Step appears most usefully in two sources: G. Hepburn Wilson's How to Dance the Modern Dances (New York, 1914) and in one of a series of flip-books published in 1914 by tobacco company Liggett & Myers to promote its cigarette brands, including Fatima and Sultan. There was an entire series of ten of these, with figures for the one-step, tango, hesitation waltz, and maxixe. These flip-books were reprinted in the 1990s, and the reprints can still be found for sale now and then.
Wilson depicts his dancers with the lady's hands raised, elbows bent, and the gentleman behind her, as shown in the illustration at left (click to enlarge). He describes the step thus:
As the lady and the gentleman two-step to the left, the gentleman is looking over the lady's left shoulder, and then in continuing the two-step movement to the right, the gentleman looks over the lady's right shoulder. This step is termed the FLIRTATION STEP.
In cigarette flip-book (letter "I" in the series), "Flirtation Steps of the Maxixe", more figures are described in the brief text than are actually shown. The text that matches the actual demonstration is:
This step is done by the man swinging to the right of partner and looking at her, then swinging to the left and again looking, still keeping up the same step, the girl turning her head first to the right, then to the left, looking at the man.
This is basically the same as Wilson's description, other than starting on the opposite side.
The actual flip-book images have the couple starting with the man almost directly behind the woman, his head over her left shoulder, almost cheek-to-cheek. The arms of both dancers are fully extended to the side. They two-step forward for a few steps, using the typical heel lead of the maxixe and a great deal of body sway toward the lead foot. They then switch to the actual Flirtation Step: as they two-step to the left, he shifts somewhat to her left, her right arm bends, and they look at each other. This is repeated in mirror image to the right. There is much less body sway during this part than there was when he was directly behind her.
Here's one of the original Sultan Cigarettes Flirtation Step flip-books in action:
A stop-motion-style animation of a Fatima Cigarettes flip-book from the collection of Richard Powers may be seen here.
I think the performance and leading of the Flirtation Step are too self-explanatory to need a count-by-count breakdown of how to do it. Briefly: start with the gentleman behind the lady, arms up or extended. From the same starting foot, both two-step forward with a heel lead and some body sway, reducing that as the gentleman starts taking slightly longer steps to switch from side to side.
How close the dancers want to be (both body-to-body and for the cheek-to-cheek initial position) depends on how comfortable they are with each other, but the arm position is awkward for the lady if the gentleman is too far behind her. This is also a move in which it really helps if the gentleman is taller and (especially if they use the extended-arms position) has slightly longer arms than the lady!
To get into position for the Flirtation Step, one can start from a standard skaters' position (left arms extended, right arms at lady's waist) and gently unfold right arms to extend or raise them. It's also possible to transition into either position directly from a closed position with left hands raised overhead and right hands behind the lady's back.
An interesting variant on the Flirtation Step is included in a booklet of descriptions of dances published by the American National Association [of] Masters of Dancing (Pittsburgh, 1915, though the dances are from their 1914 meeting). The figure is called La Coquette and is described as follows:
Lady stands in front of gentleman, and both begin with L. ft. facing line of direction.
"Pas Chasse" lady moving in front of gentleman to his L. side both raising L. hands still clasped above shoulder level, Lady looking over her R. shoulder at gentleman...counting 1 and 2 and
"Pas Chasse." lady passing to the right, looking at gentleman over her L. shoulder..counting 3 and 4 and
The overall effect is the same as in the Flirtation Step described by Wilson and shown in the flip-book, but in La Coquette the lady is doing the moving from side to side rather than the gentleman. Since this was part of a choreographed sequence, the lady would simply know what to do and when to do it, and the gentleman would know not to interfere. But it is also possible for the gentleman to gently lead the lady from side to side along with or instead of moving himself.