Schrödinger's MatrixForm: feature or bug?

Here are few notes:

Notice the second argument of underying TagBox[gridbox, Function[e, MatrixForm[e]]. It guides the interpretation from standard to input form (Ctrl+Shift+I). Since MatrixForm won't be forgotten, route back to boxes works too.

It is not documented but it seems to work like ToExpression[gridbox, StandardForm, Function[...]]. It won't work manually, ToExpression@ToBoxes@MatrixForm will return an array. Whether that is expected I don't know but I think this should be more consistent.

Here is another example (bug) showing the difference between Cell/ConvertTo items and composition of ToBoxes/Expression:

TagBox + InterpretationBox vs ConvertTo/StandardForm

Summing up:

• Ctrl+Shift+I behavior is governed by TagBox (fine but undocumented)

• ability to evaluate typeset MatrixForm comes from the fact that ToExpression strips it before parsing/evaluation. (arguable)

This is essentially a duplicate of this question:

1. When typesetting as output, MatrixForm[_?ArrayQ] will display matrix-like.
2. In an output cell, Convert To StandardForm will default to box formatting, so you go back and forth between a matrix-like state and an InputForm-like state.
3. Once you edit the cell becomes an Input cell, so now MatrixForm will use an input-form like state for Convert To StandardForm.

So what's going on with that TagBox, and how does ToExpression@ToBoxes@... produce a list? Well, normally the second argument of a TagBox, in the absence of more specific rules, is applied as the head of the expression generated from the first argument:

MakeExpression[TagBox["a", b], StandardForm] 
(* HoldComplete[b[a]] *)

However, an exception is made for heads that are members of $OutputForms, where it is ignored:

$OutputForms // Unprotect
AppendTo[$OutputForms, b]
MakeExpression[TagBox["a", b], StandardForm]
(* HoldComplete[a] *)

This is "wrapper stripping" (the behavior of functions that "affect formatting, but not evaluation") is implemented. But it would be bad if wrapper stripping occured when you did a Convert To *Form. Fortunately, that operation doesn't use MakeExpresion/MakeBoxes, but rather a dedicated function, which knows not ignore those TagBoxes. Here is the call to convert from StandardForm to InputForm:

BoxForm`ConvertForm[TagBox["a", b], StandardForm, InputForm]
(* "b[a]" *)

To be perfectly honest, the only reason I can explain this behavior is that I came across the source code for it while fixing a bug, and was able to reproduce the reason for it from the commit message. It is a bit of black magic, but given the relatively small number of questions about it for the last 24 years or so, it is good enough to get the job done.