To simplify the expression \((x-1)^3 (x+1)^3 (x-1)^3 (x+1)^3\), we can first observe that we have repeated factors. We can combine them as follows:

\[
(x-1)^3 (x+1)^3 (x-1)^3 (x+1)^3 = \left((x-1)^3\right)^2 \cdot \left((x+1)^3\right)^2
\]

Now, we can simplify further:

\[
\left((x-1)^3\right)^2 = (x-1)^6 \quad \text{and} \quad \left((x+1)^3\right)^2 = (x+1)^6
\]

Putting it all together gives us:

\[
(x-1)^6 (x+1)^6
\]

Finally, we can express this in a different way by factoring:

\[
(x-1)^6 (x+1)^6 = \left((x-1)(x+1)\right)^6
\]

The expression \((x-1)(x+1)\) simplifies to \(x^2 - 1\). Therefore, we have:

\[
\left(x^2 - 1\right)^6
\]

Thus, the simplified form of the original expression is:

\[
\boxed{(x^2 - 1)^6}
\]