The doosra, the difficult-to-bowl leg-spinner delivered with an off-spinning action, has been a key weapon for leading sub-continent slow bowlers in Test cricket, including Sri Lanka's retired world-record holder Muttiah Muralitharan.
Muralitharan claimed last month that Australia's failure to produce a top-line spinner since Shane Warne's retirement from Tests in 2007, can be attributed to the regimented attitude of the nation's coaches.
But former Test batsman Inverarity, 68, has cautioned against the clamour among Australia's spin fraternity to embrace the doosra.
"That's a question of integrity for Cricket Australia (CA)," Inverarity told a cricket luncheon in Melbourne on Friday.
"If you're going to bowl a doosra, that's how you do it," Inverarity said in fielding a question from a luncheon guest about whether the doosra constituted chucking.
"It's a serious issue. We've got to keep our integrity," he said.
"We've got to teach our bowlers to bowl properly.
"I'm all for them learning it but it has got to be within the rules, for the integrity of the game and Australia's cricket heritage.
"We should always run a measure over them to make sure they're bowling legitimately (according to the 15-degree threshold rule for elbow flexion)."
Inverarity's comments are likely to be met with scepticism in Asia, where most bowlers of the 'doosra' originate.
Pakistan offspinner Saeed Ajmal, the highest-profile bowler of the delivery today, was cleared by the International Cricket Council in 2009 after his action was thoroughly tested by an Australian University.
Australia World Twenty20 captain George Bailey returned from the World T20 in Sri Lanka this month calling for CA to look closely at coaching young spinners to bowl the doosra.
"You look at the guys who did well, the guys who have doosras and leg spinners who have real weapons in their armoury," Bailey said.
"You look at the subcontinent spinners, they use the 14 degrees in their elbow. I know it ruffles a few feathers with their actions but you need that to bowl those doosras.
"I don't think we have got those coaches over here that know how to do that. I think we have to access some of the subcontinent coaches."
Muralitharan, who retired from international cricket after Sri Lanka lost the World Cup final to India last year, finished his Test career with a record 800 wickets, 92 ahead of Warne.
Despite being branded a 'chucker' by sections of the press, particularly in Australia where he was called for bowling no-balls by controversial Umpire Darrell Hair, his action was cleared by ICC.
The 'doosra' - literally 'the second one' in Urdu - was invented by former Pakistan offspinner Saqlain Mushtaq.
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lol grapes are sour. They cannot bowl it so its matter of integrity for them. Same was said by English experts regarding lethal reverse swing in early 90s but later English fast bowlers started using it extensively although they lacked the class of Wasim and Waqar for e.g Darren Gough
Figure it out first before urging caution. Sure, just like the reverse swing it will be called an "art" once they finally learn how to do it.