I think what you want is this: 1) write points on the circle as
$z=(x,y)$ or $x+iy$, working in the algebra of complex numbers, then the unit inward
normal to the circle at point $z$ is $N=-z$, and the various (constant length) vectors pointing
inward at constant angle to $N$
are $N+aiz=(-x-ay,-y+ax)$ for any real constant $a$, or positive multiples of that.
What you called right and left of the inner normal correspond to $a$ positive
and negative respectively.
2) Write points on the 3-sphere as $q=(w,x,y,z)$ or $w+ix+jy+kz$, working in
the algebra of quaternions, then the unit inward normal to the 3-sphere at point $q$
is $N=-q$, and the constant length vectors pointing inward at constant angle to $N$ are $N+(ai+bj+ck)q$ for any real constants $a$, $b$, $c$, or positive multiples of that.
3) For the 7-sphere use the same type of construction with the octonian algebra.