First allow me to apologize, for having succumbed to the temptation to idealize form, in the reciprocating exercise described in the previous few posts. The problem is that most of us can't maintain that 45° angle while shifting weight onto a foot, not without putting undue stress on the knee. Maintaining knee alignment being far more important than maintaining any arbitrary foot position, let's start by modifying the angles.
So, starting as before, with your feet spread about 1.5 times the distance between your shoulders, toes turned outward comfortably, but not more than 45°, shift your weight onto one foot (it really doesn't matter which you start with), removing weight from the other foot. When the unweighted foot becomes light enough to pick up or slide, pull the toes in so they are only turned out about 15° from straight ahead.
Now begin to shift weight onto this foot until the other (the one you first place weight on) is light enough to pick up or slide, and point it directly away from the hip joint that connects it to your torso, so that hip, knee, ankle, and the ball of the foot are all aligned in a vertical plane.
Next, begin to move your weight back towards your unweighted foot, and in the same moment rotate that foot so that it is turned out about 15° from straight ahead, unweight the other foot and point it directly away from its hip, with the hip, knee, angle, and ball of the foot aligned in a vertical plane, as above.
Repeat, seeking the foot placement that results in no net movement, neither forward nor backward.
Now add the lead-with-the-eyes aspect back in. As you come to rest with nearly all of your weight on one foot and the other pointing straight away, imagine a soft shuffling noise coming from behind the ear on the side bearing your weight, turn your eyes towards the noise, allow your head to follow, and rotate your unweighted foot as you begin to shift your weight onto it.
Repeat until you are comfortable with this much.
Now, while at the end of a weight-shift, before beginning the next, raise the hand on the unweighted side with the fingers held straight and vertical, until the tops of the fingers are at eye level, palm turned half outward, with your forearm extended at about 30° from vertical, and pull the elbow in so that it's as close to your ribcage as you can comfortably place it. Place the other hand so that its fingertips are lightly touching the inside of your extended elbow, with its palm cupped and turned downward.
Lead with the eyes as before, then, in one motion, begin the rotation of your unweighted foot, begin moving your weight towards it, and move the hand that was touching the other elbow, first behind that elbow, then beneath it, then wiping along the arm towards the wrist, leading with the index finger. While this is happening, your weight is shifting and your arms are crossing in front of you, with the one that had been extended on the inside.
Properly coordinated, your head finishes its rotation and the foot from which you were removing weight becomes light enough to lift or slide at the same moment that the hand wiping along the other arm reaches the wrist, at which point both foot and hand rotate, the foot to pointing directly away from the hip and the hand that was doing the wiping motion into palm half-outward, fingers vertical, forearm partially extended. At the same time the other hand (on the side that your weight has shifted to) sinks to touch the elbow of your extended arm.
Continue to work on this until the two directional shifts (shifting weight left-to-right and right-to-left, while extending the arm opposite the weight-bearing leg) become mirror images of each other. Then begin to think about how to make each flow into the other so that they become a single movement. That is the real nature of a reciprocating exercise.