The Boeing 767 had departed from runway 16R at Sydney, NSW and the crew were tracking via a GUEST 2 standard instrument departure (SID) with an initial altitude restriction of 5,000 ft. The SID required the aircraft to track via the 163 radial of the Sydney very high frequency omni-directional radio range beacon (VOR) for 10NM before turning right to track for Glenfield. On first radio contact with Departures Control, the crew were instructed to climb to flight level (FL) 280 but, incorrectly, read back "to heading 280" and commenced a turn to the right when approximately 2 NM south of the runway. The air traffic controller did not realise that the crew had read back heading 280 and replied using the word "affirm". He also re-iterated the instruction to climb to FL280, which the crew correctly read back. This situation resulted in the crew believing that they had been cleared to climb to FL280 and to turn right to Glenfield, while the controller believed that the crew were climbing to FL280 via the SID track. The turn was observed on radar and, although there were several other aircraft in the general area, radar vectors by air traffic control ensured that no breakdown of separation standards occurred.