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Split the board into regions, one per dot. Each region must look identical after a 180° turn around its own dot.
Every dot on the board is the centre of exactly one region, and every square belongs to exactly one region. A region is correct when turning it 180° around its own dot leaves it looking exactly the same, and when it contains that dot. Tap a dot to select it, then drag across the squares you want to give it — whatever you paint, the square directly opposite the dot is painted too, so the shape stays symmetric on its own. Tapping a square that already belongs to the selected region removes it again. A dot can sit in the middle of one square, on the line between two, or on the corner where four meet, and that position is what fixes the shape. There is exactly one way to fill the whole board. Fill it and the clock stops; your time is your score, so lower is better.
Start at the edges and corners, not in the middle. A dot near a wall has almost nowhere to reach: the squares opposite it fall outside the board, so the region is forced into a small shape you can often finish immediately. Every square you settle that way shrinks the choices for everything nearby, and a handful of forced corners usually unlocks the whole board.
The most useful question at any square is not which dot could own it, but how many dots could. Look for squares with exactly one candidate — a square can only belong to a dot if the square directly opposite that dot is also on the board and still free — and fill those first. Squares in the middle typically have several candidates and are the last thing you should guess at, because a wrong guess there propagates its mirror across half the grid.
Remember that two dots can never share a square, and neither can a square that would leave a region disconnected. If painting a square would cut some other dot off from a pocket of empty squares, that painting is wrong even though it looks symmetric. Watching for pockets that only one dot can still reach is the single most productive habit once the easy edges are gone.
Use the automatic mirroring as a test rather than a convenience. Before you commit a square, glance at where its partner will land: if that partner sits somewhere that clearly belongs to another dot, you have your answer without trying it. Players who go slowly on the first ten squares and quickly on the rest finish far faster than players who paint fast and undo a lot.