Why does a T-tail produce a pitch-down moment in sideslip?
I've found some useful research here: E.C. Polhamus. Some factors affecting the variation of pitching moment with sideslip of
aircraft configurations. Technical report, NACA T.N. 4016, 1958.

The image shows the situation for high angles of $\beta$:


  At large angles of sideslip with swept horizontal
  tails, the loading will probably not be antisymmetrical mainly because
  of the difference in lift effectiveness of the leading and trailing portions
  of the horizontal tail caused by the difference in their effective
  sweep angles. This would result in a net lift induced on the horizontal
  tail which is a function of sideslip and tail height. This possible
  effect of tail height is illustrated in sketch 9 for large positive sideslip
  angles: 






Interesting to see is that the model has no wings, so we can rule out any causes related to wings, in case people were thinking in this direction.

I putsome more thought into it, and I drew these diagrams, which helped me understand the things being said in the paper.

The blue pluses and minuses are the resulting velocities caused by the horizontal velocity as a function of the sideslip ($V_{\beta}$).
The red distributions are the forces on the elevator as consequence of this $V_{\beta}$.

This component creates an increased velocity over the left side of the rudder, and a decreased velocity over the right side of the rudder.
If there's no sweep, the effects are equally strong, and no effect on the lift is present (denoted by the top two images)

However, if there's sweep, the effect on the left side is less strong, whereas it is stronger on the right side (indicated by the two tails in the bottom row).

In a T-tail, this stronger influence is acting on the lower side of the tail. As it is a minus (meaning a relative reduction in velocity), it will lead to a reduction in suction on the lower side, or an upward force, causing a pitch down.

In a normal tail, the minus will act over the upper surface. It causes a deceleration of the flow over the elevator, thereby reducing the lift it generates. This causes a pitch up movement.