Are poles on the inside diameter of a hollow diametrically magnetized sphere reverse that on the outside? On the inside diameter of a hollow, diametrically magnetized sphere like in the image below, are the poles reverse that of the outside, or, are the poles on the inside the same as on the outside? For example, if a smaller sphere is placed within this sphere, would it orient its poles in the same direction as this sphere (the pole on the inside of the sphere is reverse), or, would it orient its poles in the opposite direction (the poles on the inside of the sphere are same as on outside)?

 A: The problem you are running into is one of terminology, not physics.  There are actually two different quantities that go by the name "magnetic field."  There is the field ${\bf B}$ that appears in the Lorentz Force Law:  ${\bf F}=q({\bf E}+{\bf v}\times{\bf B})$ (in MKS units).  This field ${\bf B}$ is also the one that appears in Faraday's Law.  However, there is another quantity ${\bf H}=\frac{1}{\mu_{0}}{\bf B}-{\bf M}$ (where ${\bf M}$ is the magnetization–the magnetic dipole moment density inside matter) that is also frequently known as the "magnetic field."
The reason for the confusing terminology is that while ${\bf B}$ is the more fundamental field, ${\bf H}$ is easier to calculate when there are magnetic materials present, because ${\bf H}$ depends directly on the free current, which is what you control with a battery.  ${\bf H}$ also behaves more like the electric field ${\bf E}$ than ${\bf B}$ does; in particular, in the absence of free currents, both ${\bf E}$ and ${\bf H}$ have vanishing curl.  So it was thought, in the nineteenth century, that ${\bf H}$ was the more fundamental field, and ${\bf B}$ was known as the "magnetic induction" or "magnetic flux density."  Nowadays, while everyone recognizes that ${\bf B}$ is the basic field, and ${\bf H}$ is a convenient auxiliary quantity, there is still no agreement on which quantity should be called "magnetic field."
In vacuum, the difference between the ${\bf B}$ and ${\bf H}$ is unimportant; they just differ by a constant factor.  However, when magnetic materials are present, they two quantities can behave quite differently.  The north and south poles of a permanent magnet act like sources of ${\bf H}$, with the lines of ${\bf H}$ emerging from the N pole and ending at the S pole.  On the other hand, the lines of ${\bf B}$ (which is always divergenceless) form closed loops; this means that they point in opposite directions inside and outside a permanent magnet, since they much loop back around to where they started.
Here is an image of what ${\bf H}$, ${\bf B}$, and ${\bf M}$ are all doing inside a uniformly magnetized block of material.

