Do tornadoes have eyes?
Yes, if one takes the common meaning of the term "eye of the storm" to be the area of relatively low wind speed near the center of the vortex, most tornadoes can be said to have eyes. Cyclostrophic balance describes a steady-state, inviscid flow with neglected Coriolis force:

$$
\dfrac{v^2}{r} = -\dfrac{1}{\rho}\dfrac{\partial p}{\partial n}
$$  

where centripetal force balances radial pressure gradient. Here, $v$ is tangential wind speed, $r$ distance from vortex center, $\rho$ is air density, $p$ is atmospheric pressure and $n$ is the radial direction pointing inward. From here, tangential wind speed is simply:

$$
v = \sqrt{-\dfrac{r}{\rho}\dfrac{\partial p}{\partial n}}
$$

suggesting that $v\to0$ when $r\to0$. While the flow in tornadoes is highly non-stationary and subject to friction, this idealized model shows why there must exist an "eye" inside a vortex or an area of closed circulation. This "eye" may or may not be easily recognized by a hypothetical human observer inside a tornado.