Sixth Freedom: Right to Carry Traffic Between Two Foreign Countries via the Home Country
This is essentially a combination of the third and fourth freedoms, allowing an airline to connect two foreign countries through its home base.
For example, American Airlines can transport passengers, cargo, or mail between Madrid and Toronto, if it passes through the US.
Seventh Freedom: Right to Carry Traffic Between Two Foreign Countries Without Passing Through the Home Country
Allows an airline to operate a route entirely between two foreign countries, without any connection to its home country.
For example, Air France could operate a direct flight between New York and Tokyo.
Eighth Freedom: Right to Operate Air Services within a Foreign Country as Part of a Service to/from the Home Country
This is the right to operate domestic flights within a foreign country.
For example, Air Canada could operate a flight from Toronto to London and, once there, could offer flights from London to other cities within the UK.
Ninth Freedom: Right to Operate Air Services within a Foreign Country Without Any Service to/from the Home Country
Continuing with the previous example, with the Ninth Freedom, Air Canada could operate a flight entirely within the UK, without needing to connect it to a flight to or from Canada.
Both the eighth and ninth freedoms are usually subject to significant restrictions and are not frequently granted. Both are forms of cabotage, which is highly restricted by most countries, tending to protect their domestic markets and limit the capacity of foreign airlines.