This puzzle is part of the Monthly Topic Challenge #11: Now in 3D.
A typical '2D' connecting wall puzzle (as made popular by the BBC quiz show, Only Connect) involves dividing a group of 16 seemingly unconnected words into four thematic groups of four, an activity which is represented by a 4x4 grid of squares.
This puzzle takes this to another dimension!
The diagram below depicts three adjacent sides of a 4x4x4 cube, each of which is its own 4x4 connecting wall comprising the faces of 16 smaller 'cubelets'. A pool of 37 words must be divided so as to form 12 thematic groups of four. Naturally, this means that - unlike in the 2D connecting wall - in a 3D connecting wall some words must appear in 2 or (in one case) 3 different categories at once! This can best be understood by imagining that each 'cubelet' in the diagram is assigned a word, which is written on each of its visible faces.
The three interconnected connecting walls are labelled T (top), R (right) and L (left), with each category (represented by the subscript numbers 1-4) being written out in the direction of the neighbouring arrow. For example, the elements of category T1 of the top connecting wall are shaded in red and appear along the northwest-southeast diagonal. Each of these elements also appears as an entry in one of categories L1 to L4 (three of which have been shaded pink) - and, in fact, the word to be positioned in the diagram's central cubelet (common to categories T1 and L1) further appears in category R1 also! The word to be placed in the central cubelet is omitted from the list of words provided and must be deduced in order to complete the puzzle.
You are provided with 36 of the 37 words to complete this 3D connecting wall, as follows:
If you require a structure for completing the diagram, perhaps the 'least chaotic' approach is to present elements 1 to 3 of each category in alphabetical order. One word (VEIL) has been positioned for you according to this style - the other words can be inserted around it. (You may or may not find this helpful.)
TASK: Position the 36 given words in this 3D connecting wall - one to a cubelet - in order to form 12 thematic groups of size four across the three visible sides of the full cube (in the directions of the given arrows). Discover the hidden word (not provided here) which should be positioned on the diagram's central cubelet, to discover what you get when you mix the red (T1), blue (R1) and yellow (L1) connected groups!
NB This puzzle type has the potential to be much more difficult than a standard connecting wall puzzle - both to solve and to set. All efforts have been made to try and ensure that a unique solution arrangement of clearly defined categories exists.