We have already seen this reaction:

We are aware that depending on the conditions (i.e. whether this is done photochemically, or thermally) there are two different products attainable.

These results are achieved with the fact that the trans isomer is the more stable apparently being irrelevant.  Let us look at the thermally induced case first:

We must consider the HOMO of the octatriene.  Since there are three double bonds, there will be 6 pi electrons, and each orbital will be able to accommodate two electrons, then the HOMO will be Ψ3, which looked like this:

Applying this to octatriene, we can deduce that the phases of this HOMO are:

Knowing that bonding interactions are formed between lobes of the same phase, and that antibonding interactions are formed between those of opposite phases, we can deduce that there is only one way in which the molecule can react thermally:

This particular motion is called disrotatory, as the two ends of the molecule are rotating in opposite directions (clockwise and anticlockwise).

If there was a conrotatory motion, we would end up with an antibonding interaction:

Clearly, this does not result in a reaction.

We are now in a position to explain the apparently arbitrary adoption of cis geometry of the product.  We can see that a trans geometry could be achieved if the reaction underwent a conrotatory motion, but this, as we can see above, results in an antibonding interaction, and hence no reaction occurs.  The reaction which does work conversely results in the trans isomer.

However, earlier we stated that the equivalent photochemically initiated reaction results in the trans isomer.  The reason for this is that during the photochemical initiation, an electron is promoted from Ψ₃ to Ψ₄, and hence we now have to consider the HOMO as Ψ₄:

As we can see from above, with Ψ₄, the situation is reversed.

In general, we can make the following predictions: