The tool kit consisted of four parts; a crevice tool, a dusting brush, a small tool, and an adaptor. On either side of what I thought was all cleaners like the DC01 were two plastic lumps of different sizes. On the larger side, the crevice tool was stored, with either the dusting brush or small tool pushed onto the top of that.
On the other side of the cleaners, the adapter fitted to the smaller section of the plastic lump, and onto that the remaining tool, be it the dusting brush or small tool. However, this lump was tapered in size and the bottom of it was the same diameter as the lump on the other side, meaning that a tool could still be attached for storage, even without the adapter.
But as per the original message and photographs here, we can see that the adapter was never part of the original design, it would seem.