I have an incomplete understanding of this, but I will try to say what I know.

For each $c \in \mathbb{C}$, we define $Verma_c$ to be the category whose objects are Verma modules $V_{h,c}$ of central charge $c$ and highest weight $h$ for some $h \in \mathbb{C}$, and whose maps are Virasoro-module maps.  Feigin and Fuks worked out the morphisms in all of these categories (they are embeddings), and they found that the categories $Verma_c$ and $Verma_{26-c}$ are antiequivalent.  In particular, when $V_{h,c}$ has an embedded Verma submodule $V_{h+x,c}$, the corresponding module $V_{-1-h,26-c}$ embeds into $V_{-1-h-x,26-c}$.

When you consider representations of Virasoro with central charge $0 < c < 1$ and positive $h$, the corresponding objects will have central charge $25 < 26-c < 26$, and highest weight that is negative.  In particular, unitary representations don't seem to make an appearance anywhere in the complementary picture.

According to [this MathOverflow question][1], Positselski has developed the antiequivalence into a derived category statement, where modules are resolved by complexes of Vermas.  However, I have been unable to extract an explicit theorem about Virasoro from [his monograph][2].

  [1]: http://mathoverflow.net/questions/6923/complexes-of-representations-with-complementary-central-charges
  [2]: http://arxiv.org/abs/0708.3398