Hello MajorNearMiss,
The ico shere is not ideal for mirroring. A UV-Sphere or a rounded cube have natural edges at the symmetry plane, so with an axis aligned view it’s easy to remove exaclty half of it.
Not really sure if this is what you are wanting to do, but, Add your iso-sphere, then knife the project and add your mirror modifier. Here I knifed an Iso-Sphere down to a quarter then mirrored it back into the sphere.