Centering a numerical predictor around some value just amounts to reparameterization of the model. The model
\begin{equation}
y = a + b x + e
\end{equation}
is equivalent to the model
\begin{equation}
y = a' + b' (x-x_0) + e
\end{equation}
The relationship between the two parameterizations can be seen by rewriting the reparameterized model on the same form as the original model:
\begin{equation}
y = \underbrace{a'-b'x_0}_a + \underbrace{b'}_b x + e.
\end{equation}
From this we see that $a = a'-b'x_0$ so intercepts in the two models are different but since $b=b'$, the slopes are equal. So any kind of centering around some value $x_0$ does not influence the interpretation of the slope nor estimates of its standard error, confidence intervals or associated hypothesis tests.
Similar things holds true if your model includes an interaction between $x$ and some factor $i$. Then
\begin{equation}
y = a + b x + c_i + d_i x + e
\end{equation}
is equivalent to the $x_0$-centered model
\begin{equation}
y = a' + b' (x-x_0) + c'_i + d'_i (x - x_0) + e.
\end{equation}
which can be rewritten as
\begin{equation}
y = \underbrace{a' - b' x_0}_{a} + \underbrace{b'}_b x + \underbrace{c'_i - d'_i x_0}_{c_i} + \underbrace{d'_i}_{d_i} x + e.
\end{equation}
From this we see that $d_i = d'_i$ so the null hypothesis $d_1=d_2=\dots=d_k$ (no interaction in the non-centered model) is equivalent to null hypothesis $d'_1=d'_2=\dots=d'_k$ (no interaction in the $x_0$-centered model). So in terms of significance of the interaction term, mean- and median-centering is of no consequence.
Estimates of main effects of the factor $i$, $c'_i$ vs $c_i=c'_i-d'_i x_0$, will be different though and hence their significance will be different too, but you usually test for significance of the main effects before significance of interactions. And without an interaction present, the significance of a factor $i$ is not influenced by any kind of centering of $x$.
Only if you believe that a reasonable null hypothesis is that $i$ has no effect on average (for an average value of $x$) given that the interaction is present, do you need to think more carefully about this.
