Let $\mathcal P_{q,s}$ denote the parallelogram with vertices $(0,\pm\frac{1}{q}), (1,\frac{s}{q}), (-1,-\frac{s}{q})$. The function 
$$I_{q,s}(t)=2A_s(t)+2\left\lfloor\frac{t}{q}\right\rfloor+2t+1$$ 
counts the number of lattice points in the dilation $t\cdot\mathcal P_{q,s}$. In fact this is exactly the expression you get by counting the lattice points according to their x-coordinates and then summing everything up. 

Now let's apply the map $(x,y)\to (x,sx-qy)$. The parallelogram becomes the square with vertices $(0,\pm1), (\pm1, 0)$ and lattice points are sent to lattice points $(x,y)$ satisfying $y-sx= 0\pmod q$. In this setting symmetry makes it easy to see why $I_{q,s}=I_{q,s'}$ for the four bullet points. In fact, $s'=-s$ corresponds to reflecting everything on the y axis, and $s'=s^{-1}$ corresponds to reflecting everything on the line $x=y$.

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The converse is much harder and has a really fascinating story to it. First let me mention why people care about this problem. It turns out that this Ehrhart function encodes the multiplicities that $t(t+2), t\in \mathbb N$ appears as an eigenvalue of the Laplace-Beltrami operator on the <a href="https://en.wikipedia.org/wiki/Lens_space">lens space</a> $L(q,s)$. Knowing that $I(q,s)=I(q,s')$ implies that the two lens spaces $L(q,s)$ and $L(q,s')$ are _isospectral_ (have the same eigenvalues and respective multiplicities). 

What is easy to see is that the four bullet points are exactly the conditions which specify lens spaces up to _isometry_. Of course we have 
$$\text{isometric} \implies \text{homeomorphic} \implies \text{isospectral}$$
but for 3-dimensional lens spaces it turns out that the opposite implication holds as well. Already the fact that the first arrow is invertible required introducing the notion of <a href="https://en.wikipedia.org/wiki/Analytic_torsion">Reidemeister torsion</a>. From the above discussion your question is completely equivalent to the statement that isospectral lens spaces in dimension 3 are actually isometric (this is known to be false in higher dimensions). So this is some evidence that the proof will probably have to be a little difficult.

One proof is given in 
>A. Ikeda, Y. Yamamoto, "On the spectra of 3-dimensional lens spaces", Osaka J.
Math. 16(2) (1979), 447-469

for the special case where $q$ is a prime power or twice a prime power. It is proven in full generality in

> Y. Yamamoto, <a href="https://projecteuclid.org/euclid.ojm/1200772803">"On the number of lattice points in the square |x| + |y| ≤ u with a certain
congruence condition"</a>, Osaka J. Math. 17(1) (1980), 9–21

The main number theoretic ingredient in Yamomoto's proof is the fact that the numbers $\cot \frac{k\pi}{q}$ for $1\le k\le \frac{q}{2}, \gcd(k,q)=1$ are linearly independent over $\mathbb Q$. If your proof is simpler than Yamamoto's it might be worth publishing for that reason alone.