Dual
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L is an optional array.  
R is an array.  
f is an arbitrary ambivalent function.  
g is an arbitrary monadic function with an inverse. 
Introduction
The concept of the Dual operator was described in Ken Iverson's paper on Operators and Functions.
Implementation
This operator is implemented via the anonymous function:
{⍺←⍣0 ⋄ ⍵⍵⍣¯1 (⍵⍵ ⍺) ⍺⍺ ⍵⍵ ⍵}