To get rid of the reverse engineering one figures out what the HodgeDual[] is in tensor operations within an euclidean metric
In[33]:= (* (i) Hodge Dual {a,b,c} *)
Normal[LeviCivitaTensor[3] . {a, b, c}]
Out[33]= {{0, c, -b}, {-c, 0, a}, {b, -a, 0}}
replacing the Dot[] by canonical operations it reads
In[34]:= (* (ii) Hodge Dual {a,b,c} *)
(1/1!) Normal[TensorContract[TensorProduct[LeviCivitaTensor[3], {a, b, c}], {3, 4}]]
Out[34]= {{0, c, -b}, {-c, 0, a}, {b, -a, 0}}
which makes it easy to apply it twice in order to create the identity
In[35]:= (* HodgeDual HodgeDual {a,b,c} *)
(1/2!) Normal[ (* HodgeDual *)
TensorContract[TensorProduct[LeviCivitaTensor[3],(* HodgeDual *)
(1/1!) TensorContract[TensorProduct[LeviCivitaTensor[3], {a, b, c}], {3, 4}]], {{2, 4}, {3, 5}}]]
Out[35]= {a, b, c}
by the way this is in the references of the LeviCivitaTensor LeviCivitaTensor, Examples -> Applications