Mirror on a different axis

Draw a guide at the wished center. (Not necessary if the mirror axis is already in place.)

Choose the Mirror Tool.

Set the "Mirror center point" with your Alt key

Mirror your element.

Peter,

If you have 2 Anchor Points on the centre axis of the bowl (presumably one is the rightmost Anchor Point on the half circle), you can (Smart Guides being your friends so you can snap):

1) Copy the arrow in front of itself;

2) With the copy selected, switch to the Reflect Tool;

3) Click on both the Anchor Points on the centre axis of the bowl to specify that as the axis of reflexion.

If you have less than 2 Anchor Points on the centre axis of the bowl, you can you can (Smart Guides still being your friends so you can snap):

1) Copy the arrow in front of itself;

2) Select the bowl and ShiftClickDrag horizontally with the Line Segment Tool from the centre (if not shown, choose Show Center in the Attributes panel), no specific length of the line is needed;

3) With the copy arrow from 1) selected, switch to the Reflect Tool;

4) Click on the line from 2) in two places on the line (no need to Click on the end Anchor Points) to specify the line as the axis of reflexion.

If you have 1 Anchor Point on the centre axis of the bowl (presumably the rightmost Anchor Point on the half circle), you can also (Smart Guides still being your friends so you can snap):

1) Copy the arrow in front of itself;

2) ShiftClickDrag horizontally with the Line Segment Tool from an Anchor Point on the centre axis of the bowl, presumably the rightmost Anchor Point on the half circle, no specific length of the line is needed;

3) With the copy arrow from 1) selected, switch to the Reflect Tool;

4) Click on the line from 2) in two places on the line (no need to Click on the end Anchor Points) to specify the line as the axis of reflexion.

You can read more about it here,

https://helpx.adobe.com/illustrator/using/rotating-reflecting-objects.html

I'm not quite sure I understand exactly how you want to flip it, so here's how the mirror tool works: https://helpx.adobe.com/illustrator/using/rotating-reflecting-objects.html