How do I write bicross product symbols in latex?

Just combine existing symbols:

\documentclass{article}

\usepackage{amssymb}

\begin{document}

$\blacktriangleright\mathrel{\mkern-4mu}<$,
$>\mathrel{\mkern-4mu}\blacktriangleleft$,
$\blacktriangleright\joinrel\mathrel{\triangleleft}$,
$\mathrel{\triangleright}\joinrel\blacktriangleleft$

\end{document}

\joinrel is defined (robustly) as \mathrel{\mkern-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.

$>\joinrel\mathrel{\triangleleft}$ vs. $\rtimes$
$\mathrel{\triangleright}\joinrel<$ 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:

\documentclass{article}
\usepackage{mathtools,amssymb,scalerel}
\newcommand\bicrossl{%
  \mathrel{\scalerel*{\mathrel{\triangleright}\joinrel\blacktriangleleft}{x}}}
\newcommand\bicrossr{%
  \mathrel{\scalerel*{\blacktriangleright\joinrel\mathrel{\triangleleft}}{x}}}
\newcommand\biopencrossl{%
  \mathrel{\scalerel*{>\kern-.4\LMpt\joinrel\blacktriangleleft}{x}}}
\newcommand\biopencrossr{%
  \mathrel{\scalerel*{\blacktriangleright\joinrel\kern-.4\LMpt<}{x}}}
\begin{document}
$x\bicrossr y$ and $x\bicrossl y$, 

$x\biopencrossr y$ and $x\biopencrossl y$, 

$\scriptstyle x\bicrossr y$ and $\scriptstyle x\bicrossl y$, 

$\scriptstyle x\biopencrossr y$ and $\scriptstyle x\biopencrossl y$, 
\end{document}