Because that character is neither in the open symbol font nor in the star symbol
font, there is little chance to get it as "build in" symbol. But you can add it
as user defined symbol. You find ⇉ (U+21c9) for example in the lucida sans font,
which comes with the JRE.
What is the code point for your secong wish? I don't find it in the unicode tables.

Yes, there are a number of signs that are missing.

I will add here some signs often used in *chemistry*:

 1. Equilibrium:
    (aka double half arrows for chemical equilibrium)

    simple: dual arrowheads: A <----> B

    complete:
                k1
          _____________\
    Start -------------  Product
          \     k2

    where k1, k2 are the equilibrium constants (are omitted sometimes)

 2. Reaction progression:
    one of the reaction less likely / unlikely

                k1
          _____________\
    Start      --------  Product
               \   k2

    especially k2 is omitted

 See e.g. on wikipedia: http://en.wikipedia.org/wiki/Chemical_equilibrium
 (although the notations there are very basic, yet more than OOo can do)

 3. What is also missing are the 'circled-operators':
    '+', '-', 'x', '*', 'division' inside a circle

 4. Greek alphabet would be nice, too
    It would be practical to have it in the formula editor

 5. Broken arrows:
    (I did NOT find them)

    ---X--->, ---//--->, ---/---> and the double arrow versions

Well, for an image with equilibrium constants, see:
http://goldbook.iupac.org/SVG/C01480-1.png

This one uses --> arrows, but half arrows (---`) are also common and sometimes
preferred.

What I also miss in chemistry-notation is a bigger dot, e.g.:
  X{dot { }} where the dot is drawn significantly bigger
  [depicts a radical in chemistry)

@discoleo: Please have a look at the arrow-section in the Lucida Sans font, you
will find most of the arrows you need.

You get the circled operators with oplus, ominus, odot, otimes, odivides. Please
look in the "Formula Reference Tables" section in the help. Math knows more,
than available in the Selection Window. You must type it directly into the
command window.

You get greek characters for example with %delta for lower case and %DELTA for
upper case.

Really difficult are symbols, which has no unicode code point, the others you
can get, if you own a font which has glyphs for them. (You might watch the
http://www.stixfonts.org/ project.)

Sorry for the multiple posts, just found more images:

 see http://goldbook.iupac.org/SVG/F02463.png

For various other ideas, see also the complete IUPAC site:
http://goldbook.iupac.org/list_images.html

  > Please have a look at the arrow-section in the Lucida Sans font,
  > you will find most of the arrows you need.

Indeed, interesting font. Thanks. Though I still have to complain, that for
chemical annotation, you need longer arrows, so that one can write the reaction
constants.

Also, what are missing are the unequal bi-directional arrows. Please note, these
are likely not available as font-glyphs, but the math editor should handle them
as well.

  > You get greek characters for example with %delta for lower case and
  > %DELTA for upper case.

Truly, though I don't understand why the math editor window does not display the
Greek letters. It is surely easier to click the Greek 'd' than typing the whole
word '%delta'. Greek letters are so extensively used in mathematics, that I have
to rumble because this is not yet implemented.

Well, TEX has a wide range of useful additions that are currently lacking in OOo.

Just to extend the arrows-issue in chemistry, I append some TEX-elements
   Table 132: chemarrow Extensible Accents
      \autoleftarrow{abc}{def}
      \autorightarrow{abc}{def}
      \autoleftrightharpoons{abc}{def}
      \autorightleftharpoons{abc}{def}
   These symbols are all “reverse accents†in that the two arguments serve,
   respectively, as a superscript and a subscript to the arrows.
   In addition to the symbols shown above, chemarrow also provides
      \larrowfill,
      \rarrowfill,
      \leftrightharpoonsfill, and
      \rightleftharpoonsfill macros.

For an extended list of TEX-symbols, see
http://www.ctan.org/tex-archive/info/symbols/comprehensive/symbols-a4.pdf.

Indeed, the http://www.stixfonts.org/ project looks promising, too.

Adding Greek letters to selection window is another issue, but I think you need
no new one, but it is covered with issue 6535.

Labeled long arrows is issue 60591 and issue 43024.

We've got others issues about adding symbols. Therefore this issue should focus
on one problem.

To enhancement: Provide additional arrow symbols which are often used in
chemistry and provide arrow symbols which are not available as unicode character.

discoleo->regina
Thanks for the previous hints. I have added the relevant comments to those
issues, as well.

Please also close the double issue 80817.

Regarding the initial issue:
The first point is about TEX's \rightleftarrows and the 2nd point is probably
about negated arrows, see TEX:
   Table 83: AMS Negated Arrows
    \nLeftarrow
    \nLeftrightarrow
    \nRightarrow
    \nleftarrow
    \nleftrightarrow
    \nrightarrow

See on p31 of the TEX document:
http://www.ctan.org/tex-archive/info/symbols/comprehensive/symbols-a4.pdf.

As I think more thoroughly, why shouldn't OOo be able to import these TEX
symbols (TEX librarires)?

*** Issue 80617 has been marked as a duplicate of this issue. ***

Reassigned to requirements.
There are also some other symbols which are used mainly for chemical
illustration listed in this issue.

A workaround is to use Unicode characters.  Eg 
⇉  U+21C9

Example in Math:
a "⇉" b

You have to use quotes, otherwise the double arrow is italic!

Or you could use the Wikipedia trick :-/ as I've stated in issue 60591 comment #5.