How can an intense solar storm harm the power grid?
Picture a large power line, with its distribution towers. There are long, open loops formed by the conductors, and between each conductor and the earth ground. This has a huge area, probably measured in tens or hundreds of thousands of square meters. So a very-large-scale changing magnetic field induces a huge change in flux around the loop, causing large voltages and currents to result.
Presumably the phase conductors are rotated every once in a while, so it's sort of a large-scale twisted triple, and that could be used to reduce the effective flux area, but there's no way to do that between the 3 phases and ground, and still keep the conductors above ground, so the ground-to-phase loop areas are much larger.
As for your question about the neutral conductors, there was a good article in the Feb 2012 issue of IEEE Spectrum about solar storms, including this section, which is very relevant to your question: (GIC = ground-induced current)

Fortunately, protecting against the space-weather threat should be neither expensive nor difficult. Almost all modern power grids use a three-phase design, in which each of three lines carries an alternating current whose phase is separated from the other two by a third of a cycle.
The three-phase transformers used in this scheme are directly grounded through their neutrals. The ground is assumed to be an infinite sink that can absorb any brief, large fault current and at the same time keep the voltage across the grid from spiking and damaging equipment. In retrospect, that design is flawed: During a geomagnetic storm, the sink becomes a source of GICs, flowing into the grid from the ground.
Preventing the inflow of GICs into the grid through the neutral-to-ground connection is the best long-term solution. One idea is to install capacitors at the neutral-to-ground juncture, which would block the inflow of GICs or any other direct current but otherwise allow the continuous flow of small AC currents that are common from the neutral to the ground. The challenge is devising a way to automatically bypass the capacitor in the event of an actual fault and allow a large AC current—say, more than 20 kiloamperes—to flow to ground from the neutral.