The direct answer to you question is: No, you may not in general identify $\phi=-\pi/2$ with $\phi=\pi/2$. The points to not correspond to the same physical conditions. The exception is if you are in fact treating a two-dimensional particle (an ellipse) in a two-dimensional world. But as the question stands that does not seem to be the case.
Now, I can clarify a little.
a) The equation in the question is valid for ellipsoids of revolution, spheroids. The angle $\phi$ is the azimuthal angle in a spherical coordinate system describing the axis of revolution of the spheroid.
b) The original question arises because of the inverse tangent in the solution. The original derivation of the equation, due to <PERSON> (1922) DOI link (pdf), instead states (Eq. 48): $$ \tan\phi=\frac{a}{b}\tan\frac{\kappa abt}{a^2+b^2}. 359
$$ This leaves no ambiguity. The function $\phi(t)$ is a solution of the equation of motion (Eq. 47): $$ (a^2+b^2)\dot\phi = \kappa(a^2\cos^2\phi + b^2\sin^2\phi). $$ If this equation is integrated numerically one finds that $\phi(t)$ grows monotonically (for physical values of the parameters).
c) Depending on what application you have in mind, you may consider working directly with the cartesian components of the unit vector $\mathbf n(t)$, instead of the spherical parametrization. The corresponding equation of motion is $$ \dot{\mathbf n}=\mathbb O\mathbf n + \Lambda(\mathbb S\mathbf n-\mathbf n \mathbf n^T\mathbb S \mathbf n) $$ where the constant flow-gradient matrices $\mathbb O$ and $\mathbb S$ represent the flow, and the constant $\Lambda$ represents the shape of the particle (related to $r$ in question). This equation is, perhaps surprisingly, exactly solvable by a simple matrix exponential: $$ \mathbf n(t)=\frac{\exp[(\mathbb O + \Lambda \mathbb S)t]\mathbb n(0)}{|\exp[(\mathbb O + \Lambda \mathbb S)t]\mathbb n(0)|} $$
I recently wrote an introduction to this subject and the equations quoted. It is available here (free full text). You may be particularly interested in Sec 2.3 (p.23) and Appendix A for the generalization to triaxial ellipsoids. 804