Based on Gavin Wolpert's webinar "Minor Suit Opening Structure"
Slightly different opener's rebids: bridgebum.com

Inverted minors flip the traditional meaning of single and double minor-suit raises. Playing inverted minors, a single raise of a 1♣/1♦ opening bid is strong and a jump raise is weak.

The Weak Raise (1♣-3♣ or 1♦-3♦)

The requirements for a weak jump raise are

• 0-6 point
• At least 5-card support
• No 4-card major

♠J93 ♥874 ♦52 ♣KQ732
1♣-3♣

♠975 ♥2 ♦Q10987 ♣10754
1♦-3♦

The Strong Raise (1♣-2♣ or 1♦-2♦)

The single raise is forcing for one round. The requirements are:

• 10+ points
• At least 4-card support
• No 4-card major

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♠J103 ♥9 ♦AQ75 ♣A10643
1♣-2♣
Your 2♣ bid shows 4+ card support and 10+ points.

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♠976 ♥A109 ♦KQ875 ♣AK
1♦-2♦
There is no upper point count limit for a strong raise.

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♠J98 ♥Q76 ♦A65 ♣K853
1♣-1NT
1NT with 4-3-3-3 shape is more descriptive than an inverted raise.

Opener's rebids

Main goal: show stoppers on the road to 3NT.

1♣-2♣

• 2♦/2♥/2♠ Stopper showing - bid your lowest stopper
• 2NT All suits stopped - forcing
• 3♣ Minimum - not worth showing a stopper
• 3♦/3♥/3♠ Jump = Splinter

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South: ♠AK86 ♥43 ♦QJ9 ♣A1042

When we show our stoppers, any suit we bypass should let partner know about a potential problem for NT.
Here we show our diamond stopper hoping partner will bid hearts or NT at their next turn.
Correct bid: 2♦

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♠Q3 ♥65 ♦KQ543 ♣AJ87

Our first few bids after inverted minors all show stoppers, and we deny a stopper in any suit we bypass.
We have a serious heart problem for NT. Our hand was close to a game force before, but now we see that the game we would need to play would be 5♦.
We should sign off in 3♦. Partner can still have a very strong hand and proceed, but we need to slow the auction down.
Correct bid: 3♦

In Competition

In competition, inverted minors are off for most partnerships. A single raise is weak with 4-card support, and a jump raise is weak with 5+ card support.
If the opponents have doubled for takeout over 1♣/♦, then responder can use the Jordan 2NT convention to show a limit raise or better.
If the opponents have made a natural suit overcall over 1♣/♦, then responder can cuebid to show a limit raise or better.

Other Considerations

Because inverted minors use 0-6 points for weak raises and 10+ points for strong ones, responder will sometimes be stuck for a bid with 7-9 points. These hands, which would traditionally make a single raise, can no longer do so. The typical approach is to respond 1NT even with unbalanced shape.

♠92 ♥87 ♦AQ75 ♣J10732
1♦-1NT
Hand doesn't qualify for anything else.