Cheesemonger's puzzle

^{An entry in Fortnightly Topic Challenge #45: Flags}

The cheesemonger wants to arrange the wheels of cheese on display. The display case has two shelves with space for three wheels of cheese on each. He has six wheels of different kinds of cheese to lay out on the shelves and some very specific ideas about their layout.

1. Cheddar has a 45° wedge cut out of it; Parmesan has exactly half of the wheel left; all other cheeses (Brie, Camembert, Emmental and Swiss) have 135° wedges cut out of them. The cheese left in each wheel forms one continuous piece.
2. All cheeses on the bottom shelf have 7 or more letters in their names.
3. The cheeses on the top shelf are placed in order of increasing name length.
4. The cheeses on the bottom shelf are placed in alphabetical order.
5. The sums of angle sizes of wedges cut out of cheeses in the left and the middle columns are equal.
6. Each cheese in the bottom row has a vertical cut.
7. Both cuts on Swiss cheese point at another cheese; for all other cheeses, exactly one of the cuts points at another cheese.
8. The cuts on each cheese with a double letter in its name point to exactly one other cheese with the same property.
9. If you start at Parmesan and follow the cuts on cheeses as they point to other cheeses, you can visit every cheese once and end at cheddar which you’ll visit for the second time.
10. No two cuts in the left column point in the same direction.

Can you figure out how to lay out the cheese and what is today’s special in the shop?