Prime Coincidence

$\displaystyle \begin{aligned} \color{#3D99F6} 2 & \Longrightarrow {\color{#3D99F6} 2}{\color{#D61F06}2} = 2 \times 11 \\ \color{#3D99F6} 67 & \Longrightarrow {\color{#3D99F6}67}{\color{#D61F06}76} = 2 \times 3388 \\ \color{#3D99F6} 479 & \Longrightarrow {\color{#3D99F6}{479}}{\color{#D61F06}{974}} = 2 \times 239987 \\ \color{#3D99F6} 8123 & \Longrightarrow {\color{#3D99F6}{8123}}{\color{#D61F06}{3218}} = 2 \times 40616609 \\ \color{#3D99F6} 56209 & \Longrightarrow {\color{#3D99F6}{56209}}{\color{#D61F06}{90265}} = 5 \times 1124198053 \\ \color{#3D99F6} 999007 & \Longrightarrow {\color{#3D99F6}{999007}}{\color{#D61F06}{700999}} = 7 \times 142715385857 \end{aligned}$

You have a machine that does some very interesting prime arithmetic. The machine takes in a prime number, reverses its digits, and attaches the resulting number to the prime input to form a new number.

It seems that for every prime number shown above (in the far left column), the machine produces a composite number (as implied by the right side of the equation).

Does there exist a prime for which the machine will produce another prime?