1. The problem states that the function $f(x) = \frac{1}{6} \left(\frac{2}{5}\right)^x$ is reflected across the y-axis to create the function $g(x)$. We need to find which ordered pair is on $g(x)$.\n\n2. Reflection across the y-axis means replacing $x$ by $-x$ in the function. So, \n$$g(x) = f(-x) = \frac{1}{6} \left(\frac{2}{5}\right)^{-x}.$$\n\n3. Using the property of exponents, $a^{-x} = \frac{1}{a^x}$, we get \n$$g(x) = \frac{1}{6} \cdot \frac{1}{\left(\frac{2}{5}\right)^x} = \frac{1}{6} \cdot \left(\frac{5}{2}\right)^x.$$\n\n4. Now, we check each ordered pair $(x,y)$ to see if $y = g(x)$.\n\n- For $x = -3$, \n$$g(-3) = \frac{1}{6} \left(\frac{5}{2}\right)^{-3} = \frac{1}{6} \cdot \left(\frac{2}{5}\right)^3 = \frac{1}{6} \cdot \frac{8}{125} = \frac{8}{750} = \frac{4}{375}.$$\nThis matches the pair $[-3, \frac{4}{375}]$.\n\n- For $x = -2$, \n$$g(-2) = \frac{1}{6} \left(\frac{5}{2}\right)^{-2} = \frac{1}{6} \cdot \left(\frac{2}{5}\right)^2 = \frac{1}{6} \cdot \frac{4}{25} = \frac{4}{150} = \frac{2}{75},$$\nwhich does not match $\frac{25}{24}$.\n\n- For $x = 2$, \n$$g(2) = \frac{1}{6} \left(\frac{5}{2}\right)^2 = \frac{1}{6} \cdot \frac{25}{4} = \frac{25}{24},$$\nwhich does not match $\frac{2}{75}$.\n\n- For $x = 3$, \n$$g(3) = \frac{1}{6} \left(\frac{5}{2}\right)^3 = \frac{1}{6} \cdot \frac{125}{8} = \frac{125}{48},$$\nwhich does not match $-\frac{125}{48}$.\n\n5. Therefore, the ordered pair on $g(x)$ is $\boxed{\left[-3, \frac{4}{375}\right]}$.