+
+
+-- | Rotate a cardinal direction counter-clockwise about the y-axis.
+ccwy :: Cardinal -> Cardinal
+ccwy F = D
+ccwy B = T
+ccwy L = L
+ccwy R = R
+ccwy D = B
+ccwy T = F
+ccwy FL = LD
+ccwy FR = RD
+ccwy FD = BD
+ccwy FT = FD
+ccwy BL = LT
+ccwy BR = RT
+ccwy BD = BT
+ccwy BT = FT
+ccwy LD = BL
+ccwy LT = FL
+ccwy RD = BR
+ccwy RT = FR
+ccwy FLD = BLD
+ccwy FLT = FLD
+ccwy FRD = BRD
+ccwy FRT = FRD
+ccwy BLD = BLT
+ccwy BLT = FLT
+ccwy BRD = BRT
+ccwy BRT = FRT
+ccwy I = I
+ccwy (Scalar s) = (Scalar s)
+ccwy (Sum c0 c1) = Sum (ccwy c0) (ccwy c1)
+ccwy (Difference c0 c1) = Difference (ccwy c0) (ccwy c1)
+ccwy (Product c0 c1) = Product (ccwy c0) (ccwy c1)
+ccwy (Quotient c0 c1) = Quotient (ccwy c0) (ccwy c1)
+
+-- | Rotate a cardinal direction clockwise about the y-axis.
+cwy :: Cardinal -> Cardinal
+cwy = ccwy . ccwy . ccwy
+
+
+-- | Rotate a cardinal direction counter-clockwise about the z-axis.
+ccwz :: Cardinal -> Cardinal
+ccwz F = L
+ccwz B = R
+ccwz L = B
+ccwz R = F
+ccwz D = D
+ccwz T = T
+ccwz FL = BL
+ccwz FR = FL
+ccwz FD = LD
+ccwz FT = LT
+ccwz BL = BR
+ccwz BR = FR
+ccwz BD = RD
+ccwz BT = RT
+ccwz LD = BD
+ccwz LT = BT
+ccwz RD = FD
+ccwz RT = FT
+ccwz FLD = BLD
+ccwz FLT = BLT
+ccwz FRD = FLD
+ccwz FRT = FLT
+ccwz BLD = BRD
+ccwz BLT = BRT
+ccwz BRD = FRD
+ccwz BRT = FRT
+ccwz I = I
+ccwz (Scalar s) = (Scalar s)
+ccwz (Sum c0 c1) = Sum (ccwz c0) (ccwz c1)
+ccwz (Difference c0 c1) = Difference (ccwz c0) (ccwz c1)
+ccwz (Product c0 c1) = Product (ccwz c0) (ccwz c1)
+ccwz (Quotient c0 c1) = Quotient (ccwz c0) (ccwz c1)
+
+-- | Rotate a cardinal direction clockwise about the z-axis.
+cwz :: Cardinal -> Cardinal
+cwz = ccwz . ccwz . ccwz