Let $\Lambda(s)=\pi^{-s/2} \Gamma\left(\frac{s}{2}\right)\zeta(s).$ Then $\Lambda\left(\frac{1}{2} + s\right) = \Lambda\left(\frac{1}{2} - s\right)$ constitutes the functional equation for the Riemann $\zeta$ function. The presence of the product $\Gamma\left(\frac{s}{2}\right) \zeta(s)$ is perfectly understandable, inasmuch as the poles of the former coincide with the (trivial) zeroes of the latter; as is also the symmetry with regard to $\frac{1}{2}$ since this value stands midway between the two poles of $\Lambda$. 


What poses **serious difficulties** from an intuitive perspective, however, is the presence of $\pi^{-s/2} \zeta(s)$ instead of the expected $\pi^{-s} \zeta(s)$ given the fact that $\zeta(2k)$ always possesses a known closed form in terms of $\pi^{2k}$ rather than merely $\pi^k$ for integer values of the argument $k$. One can, of course, always write $\Lambda(s) = \Gamma\left(\frac{1}{2}\right)^{-s} \Gamma\left(\frac{s}{2}\right) \zeta(s)$ but, for all its niceness, the latter appears somewhat contrived, inasmuch as the power of $\pi$ is clearly a contribution of Riemann's $\zeta$ rather than Euler's $\Gamma$ function.



Intuition behind the Riemann $\zeta$ functional equation
Your intuition breaks down because $\zeta(1-2k)$ has a closed form in terms of Bernoulli numbers, but no powers of $\pi$ at all. This was known to Euler (via Abel summation as the series is of course divergent). To get to something that might look symmmetric in powers of $\pi$, you need to move "half" of the $\pi$ to the other side of the equation.


