Trjtdtk

There are commands for the top two symbols ltimes and rtimes, however I have not been able to find commands for the other 4 symbols. Is there a simple way that I could create commands for these symbols?

Just combine existing symbols:

documentclassarticle

usepackageamssymb

begindocument

$blacktrianglerightmathrelmkern-4mu<$,
$>mathrelmkern-4mublacktriangleleft$,
$blacktrianglerightjoinrelmathreltriangleleft$,
$mathreltrianglerightjoinrelblacktriangleleft$

enddocument

joinrel is defined (robustly) as mathrelmkern-3mu. It's enough for the last two symbols; for the first two a slighlty larger value of 4mu looks better to me.

As a matter of fact, ltimes and rtimes do not yield the "unsymmetric" symbols in your picture. They can be similarly obtained joining </ > with triangleleft/triangleright.

$>joinrelmathreltriangleleft$ vs. $rtimes$
$mathreltrianglerightjoinrel<$ vs. $ltimes$

My fantasy isn't rich enough to come up with names for all these ;-)

This takes campa's answer (+1) and makes an enhancement/alteration: it scales the result downward to occupy the same vertical footprint as the letter x.

Like campa's result, it works across math styles.

The MWE:

documentclassarticle
usepackagemathtools,amssymb,scalerel
newcommandbicrossl%
 mathrelscalerel*mathreltrianglerightjoinrelblacktriangleleftx
newcommandbicrossr%
 mathrelscalerel*blacktrianglerightjoinrelmathreltriangleleftx
newcommandbiopencrossl%
 mathrelscalerel*>kern-.4LMptjoinrelblacktriangleleftx
newcommandbiopencrossr%
 mathrelscalerel*blacktrianglerightjoinrelkern-.4LMpt<x
begindocument
$xbicrossr y$ and $xbicrossl y$, 

$xbiopencrossr y$ and $xbiopencrossl y$, 

$scriptstyle xbicrossr y$ and $scriptstyle xbicrossl y$, 

$scriptstyle xbiopencrossr y$ and $scriptstyle xbiopencrossl y$, 
enddocument