Changing the orientation of the capactior in a simple integrating op amp
No, you will get the same result in both cases, which is 0.  You can see this is true by definition since the output is grounded.

Even with the output not grounded, flipping the capacitor won't make any difference.  The polarity of a capacitor does not decide the polarity of the voltage applied to it.  It only decides which polarity of applied voltage will cause the capacitor to blow up.  This is assuming a type of cap to which polarity matters, such as electrolytic or tantalum.

In your circuit, the right side of the capacitor will always be at or lower than the left side due to the fact that only positive voltage is applied into the integrator.  This particular integrator inverts as a side effect of its topology.  Note that this circuit won't work unless the opamp has a negative supply.  Without a negative supply, the opamp can't swing below ground and therefore can't make the expected output voltage.