odd shift

double  xshift[2] = {rp.Shift.x * m_st.Zoom, (rp.Shift.x + rp.OddShift.x) * m_st.Zoom};
double  yshift[2] = {rp.Shift.y * m_st.Zoom, (rp.Shift.y + rp.OddShift.y) * m_st.Zoom};

Notice that the calculation is NOT corrected for radius. This causes the effect to decrease with distance from the origin. Correcting for radius gives a very different and less pleasing effect. The display becomes too busy, and at extreme values, the image is distorted into a narrow cylinder. The first objection could possibly be addressed by limiting the size of the ring list.

To correct the above for radius:
double  xshift[2] = {rp.Shift.x * m_st.Zoom, (rp.Shift.x + rp.OddShift.x * rp.Steps) * m_st.Zoom};
double  yshift[2] = {rp.Shift.y * m_st.Zoom, (rp.Shift.y + rp.OddShift.y * rp.Steps) * m_st.Zoom};

Some possible names for "odd shift":

Shear: incorrect usage
Extrude: incorrect usage
Tilt: too vague
Tunnel: too vague, also other parameters can produce this effect
Relief: too vague
Stamen: too obscure
Center Fold: misleading, the new version corrects for radius, see below
Splay: means legs spread at various angles, not various lengths
Deform: vague, not a noun, misleading negative connotations
Bias: misleading connotation of unfairness
Fold: vague, possibly misleading
Asymmetry: vague, too long a word

10/29/05 corrected for radius but also centered, much better

DPOINT  splay = {rp.Splay.x * rp.Steps, rp.Splay.y * rp.Steps};
double  xshift[2] = {
    (rp.Shift.x - splay.x) * m_st.Zoom, 
    (rp.Shift.x + splay.x) * m_st.Zoom};
double  yshift[2] = {
    (rp.Shift.y - splay.y) * m_st.Zoom, 
    (rp.Shift.y + splay.y) * m_st.Zoom};

Controls should be polar as with Skew: Splay Radius, Splay Angle

Some interesting interdependent terms:
Contort: Twist, wrench, or bend severely out of shape
Pinch: To press, squeeze, or bind painfully
Buckle: Bending, warping, or crumpling; a bend or bulge
Rumple: An irregular or untidy crease
Crumple: To crush together or press into wrinkles; rumple
Crease: A line made by pressing, folding, or wrinkling
Wrinkle: A small furrow, ridge, or crease on a normally smooth surface
Pucker: To gather into small wrinkles or folds

And the winner as of 12/17/05 is:
PUCKER

totally cool demo: default patch, ring spacing 11.192, pucker radius -.406, pucker angle -65.88 / ramp up / 180 / .08, master speed 222

another one: patch_051028050234, rings = 75, pucker rad = .268, angle = ramp up / 180 / .5, speed = 495

optimized 12/19/05 (two less Zoom multiplications):

DPOINT  pucker = {rp.Pucker.x * steps, rp.Pucker.y * steps};
DPOINT  shift = {rp.Shift.x * m_st.Zoom, rp.Shift.y * m_st.Zoom};
double  xshift[2] = {shift.x - pucker.x, shift.x + pucker.x};
double  yshift[2] = {shift.y - pucker.y, shift.y + pucker.y};