Does Barro understand Barro?

Clive Crook writes,

Thanks, Robert, especially for that last clarification. I have to admit that I haven't been giving much thought to the scenario in which a deficit-financed stimulus has no effect (under Ricardian equivalence) on aggregate demand and yet still has a big multiplier. I will have to think about that one...

I have always thought about Barro's Ricardian equivalence as treating any government deficit as if it were financed by taxes. That is, when people see the government borrow money, they anticipate a future taxes, and they adjust their spending exactly as if the taxes were being levied today.

So, we describe fiscal policy by saying "____ is financed by a tax increase."  So, if the government tries a tax cut, we say, "a tax cut financed by a tax increase."  I see that as obviously self-canceling, with a multiplier of zero.  Because Barro now implies otherwise, Clive Crook is confused, and so am I. 

On the other hand an increase in government spending that is financed by borrowing should, in a Barro world, be treated as an increase in government spending financed by an increase in taxes. Thus, the multiplier should equal the so-called "balanced budget multiplier,"   which can be greater than zero. 

Barro himself writes (as quoted by Crook):

The multiplier has to do with how a change in aggregate demand affects output.

I think of the multiplier as the product of two factors--how much the policy affects aggregate demand and how much the change in aggregate demand affects output.  I imagine that Crook is thinking the same thing, in which case a tax cut in a Barro world should have a mulitiplier of zero.