dg.differential geometry - When are fiber bundles reversible?

My question, in its most general form is this:

Given a fiber bundle $Frightarrow Erightarrow B$, when is there a fiber bundle $Brightarrow Erightarrow F$?

Here, F,E, and B can lie in whichever category you wish, but I'm mostly interested in the case where all 3 are smooth closed manifolds.

Now, I realize that the initial answer is "unless E is a product, essentially never", so here is a more focused question (with background).

I've been studying a certain class of free actions of the 3-torus $T^3$ on $S^3times S^3times S^3 = (S^3)^3$. For each of these actions, by quotienting out by various subtori, I can show that the orbit space $E=(S^3)^3/T^3$ simultaneously fits into 2 fiber bundles:

$$S^2rightarrow E rightarrow S^2times S^2$$ and $$S^2times S^2rightarrow Erightarrow S^2$$ where the structure group for both bundles is $S^1$.

(In fact, the class of actions also gives rise to examples where either $S^2times S^2$ can independently be replaced with $mathbb{C}P^2sharp -mathbb{C}P^2$, the unique nontrivial $S^2$ bundle over $S^2$.)

By computing characteristic classes for (the tangent bundle to) E, I know that for an infinite sublcass of the actions I'm looking at, E is not homotopy equivalent to $S^2times S^2times S^2$, and each of the E are pairwise nondiffeomorphic.

I suspect the reason I could find so many E which fit into "reversible" fiber bundles is strongly related with the fact that the fiber and base are so closely related.

And so, I ask

For fixed manifold M, what is the relationship between bundles $Xrightarrow Erightarrow M$ and $Mrightarrow E'rightarrow X$ where $X$ is some $M$ bundle over $M$?

And just in case there is no general relationship,

Is there a reason I should have expected there to be a relationship in my examples, even though in general there isn't?

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