Let's look at the following presentation: $$ \Delta^*(p,q,r;s/2)=\langle a,b,c\mid a^2=b^2=c^2=(ab)^p=(bc)^q=(ca)^r=((abc)^2)^{s/2}=1\rangle $$ This is a presentation of a special triangle group $\Delta^*(p,q,r; s/2)$.
Focusing on the corresponding index-two subgroup of $\Delta^*$ (Von Dyck group), we get $$ \Delta_0^*(p,q,r; s/2)=\langle x,y,z\mid x^p=y^q=z^r=xyz=(xzy)^{s/2}=1\rangle , $$ where $x=ab, y=bc, z=ca$ (we see $xyz=ab\;bc\;ca=1$) and $xzy=ab\; ca\; bc=(abc)^2$. It is said, that this means that $\Delta^*_0(p,q,r;s/2)$ is a discrete group consisting of orientation-preserving isometries of the hyperbolic plane. I think this relates e.g. to the regular triangles-tilings of the hyberbolic plane (correct me if I'm wrong)...
Does $(xzy)^{s/2}$ preserve some special kind of property like orientation?
The presentation in question are motivated by this and that...
As far as I can tell your $\Delta^*(p,q,r;s/2)$ isn't a triangle group, it is a quotient of the triangle group $\Delta(p,q,r)$ by adding the relation $((abc)^2)^{s/2}=1$ (It could secretly be a triangle group but I doubt it). You could say it is some "generalized triangle group" if you want but then the connection to acting on hyperbolic plane etc. starts to break.
For now just look $\Delta_0(p,q,r)$ von Dyck group with respect to the same generators you give. Every element in $\Delta_0$ preserves orientation, almost by definition, which includes $xzy$ and any power of that element. Now remember the group you describe is a quotient of the above group and you set $(xzy)^{s/2}=1$, so that element does nothing to any space you act on. If you want you can think of as orientation preserving as it "preserves" everything
Generally, if a group $G$ acts on a space $X$ then quotient groups will not, necessarily, have a natural action on $X$. As a simple example, $\mathbb{Z}$ acts on $\mathbb{R}$ by translation or you can have $\mathbb{Z}$ act on the hyperbolic plane by choosing a hyperbolic or parabolic isometry. Any action of $\mathbb{Z/7Z}$, a quotient group of $\mathbb{Z}$, on $\mathbb{R}$ must fix a point and in this case there is no nontrivial action. Similarly $\mathbb{Z/7Z}$ would have to fix a point on the hyperbolic plane, so it could be a rotation of order seven but a rotation doesn't really have anything to do with original action. Note that $\mathbb{Z}$ in the above cases does not fix any point in $\mathbb R$ or the hyperbolic plane. With that being said I don't think $\Delta_0^*$ generally has (natural) actions coming from the group $\Delta_0$.