The Leadered Tags are really a 'Schedule Tag' that was inherited from Architectural Desktop days.
The 'Schedule Tag' is using leader functionality from a dimension style. If you manually draw a leader with the stock Leader command and do not put text on it, the land or hook goes away... The 'Schedule Tag' use this 'textless' leader. The 'Schedule Tag' itself, is actually an MvBlock.
We did tags for our pipe... but we found we had to make two different MvBlocks... one for the right handed case and one for the left handed case. This required two different AutoCAD blocks one with insertion point on the left and one with the insertion point on the right. If you get the wrong MvBlock... say the one for the right hand MvBlock on a left handed leader you can just swap out the right handed MvBlock for the lefted handed one.
Joshua is partly correct... if the tag you are using is really not a tag but a Label Curve - these do align with the object as Joshua said... changing the angle with an Label Curve is very convoluted as the angle is in relationship to the pipe's internal coordinate system.
I've not found a way to have Label Curve use a Leader, although I think it may be possible with a Leader Anchor - although I doubt it.