Limiting thickness as extra dimension of Möbius band to make it orientable By incorporating thickness of Möbius band  i.e, by adding one more dimensiom as a valid dimension it is possible to orient an otherwise un-orientable Möbius band.
That is to say by gluing four un-orientable Möbius bands along their edges I obtain a single orientable torus. That is, such a four Möbius bands union is homeomorphic to the full Clifford torus.
Alternately choosing a low mesh ( 120,3 for u,v  latitude and longitude ) for parameterization of a closed geodesic in Mathematica display option in above images I made to obtain back four un-orientable Möbius bands. 
ThickMoebiusBand
Non-orientability of Möbius band seems to me to be an artifact of unbounded range of one of the defining parameters.
Is this view correct?  If not, by what change can it be accomplished ? Is incorporating orientability into the Möbius band impossible by any means?
EDIT 1:
Questions arising are: 
1) How is Orientability defined? Is there a topological equation describing it?
2) In view of the above definition (constitutive or otherwise), under what conditions of gluing/surgery is Orientability said to be gained or lost for  assembled/disjunct parts?
 A: "By incorporating thickness of Möbius band i.e, by adding one more dimensiom as a valid dimension it is possible to orient an otherwise un-orientable Möbius band."
Yes, you started with a non-orientable surface and thickened it to an orientable $3$-manifold.
"That is to say by gluing four un-orientable Möbius bands along their edges I obtain a single orientable torus. That is, such a four Möbius bands union is homeomorphic to the full Clifford torus."
You can't obtain an orientable surface by gluing non-orientable things together. Any orientation-reversing path in a piece will still be there in the whole. Maybe you meant gluing four "thickenings" of Möbius bands.
"Non-orientability of Möbius band seems to me to be an artifact of unbounded range of one of the defining parameters."
It has nothing do to with an unbounded range of any defining parameters. The Möbius strip can be parameterized with bounded ranges.
"How is Orientability defined? Is there a topological equation describing it?"
A loop in a manifold is said to reverse orientation if, when you start with a right-handed coordinate system and traverse the loop, carrying the coordinate system continuously with you, your coordinate system turns into a left-handed one. On a surface, this means clockwise will turn into counterclockwise, and on a 3-manifold, it means that if you physically walked along the path, your heart would end up being on the right side of your body rather than the left. (As viewed by those who stayed behind.) A manifold is said to be orientable if there are no such orientation reversing loops.
"under what conditions of gluing/surgery is Orientability said to be gained or lost for assembled/disjunct parts?"
As I mentioned earlier, and orientation reversing path will remain when you glue pieces of an n-manifolds to get a larger n-manifold. So you can never make an orientable n-manifold out of nonorientable n-manifold pieces. In your example, you first thickened the piece, and that's the stage at which non-orientable turned into orientable, not the gluing stage.
A: $\newcommand{\Reals}{\mathbf{R}}$This is arguably more a comment on the question than an answer, but it's too long for a comment, and requires an image.
Careful inspection of your "thickened Möbius band" shows the boundary is not four Möbius strips, but two orientable double covers of the Möbius strip: One red, one blue, and each admitting a nowhere-vanishing continuous normal vector field pointing "outward" from the solid.

The model pictured above, which may be viewed as "the layer of paint if you paint the entire exposed surface of a Möbius strip", was made by:


*

*Gluing two double-ply paper strips using white and color paper;

*Placing the strips together so the white surfaces face each other (giving, effectively, a single four-ply strip);

*Giving this strip a half-twist as if forming an ordinary Möbius strip;

*Attaching the "corresponding two-ply layers at the joint."
Alternatively, imagine cutting an ordinary Möbius strip in half by slicing the thickness of the paper to obtain two parallel sheets. The result still looks superficially like a Möbius strip, but the new surface is two-sided: You cannot pass from the face of the original strip (colored paper) to the cut face (white paper) by moving only along the surface of the strip.
Here are a couple of useful formalizations that produce the preceding model from the Möbius strip:


*

*Let $M$ be a Möbius strip embedded in $\Reals^{3}$, and let $\nu$ be the normal bundle of $M$, equipped with the bundle metric coming from the embedding. There is no continuous, non-vanishing section of $\nu$, but there is a "continuous, non-vanishing, two-valued section", which for a sufficiently small real number $t > 0$, associates to each point $p$ of $M$ the two points lying on the normal line through $p$ and at distance $t$ from $p$. The image of this multi-section is not a Möbius strip, but its oriented double cover.

*At each point of a connected, smooth manifold $M$, there are precisely two local orientations, lying in the top exterior power of the tangent bundle of $M$. Pick a point $p$ arbitrarily and one of the two orientations $O_{p}$ at $p$. Now perform the analog of analytic continuation: In the sheaf of local orientations, consider a maximal connected set containing $O_{P}$. (More concretely, extend $O_{p}$ to an open set $U$; then for each open set $V$ such that $U \cap V$ is connected, extend the section by continuity to $U \cup V$. Do this over all chains of open sets in $M$.) The resulting multi-section is, again, a double-cover of $M$, and is connected precisely when $M$ is non-orientable.
These descriptions should help clarify why thickening a Möbius strip does not give two Möbius strips, but instead "immediately" yields an oriented double cover.
