Whether capturable or mobile, obstacles should be thought of as neutral pieces that do not capture. If they are neither capturable nor mobile, then they are just features of the terrain like holes in the board and shouldn't properly be thought of as obstacles. If they can be moved by only one player and not the other, they would count as part of that player's forces rather than as obstacles. Likewise, if they could be captured by only one player, they could be considered part of the other player's forces in some sense. If they could capture other pieces, they would be equal-opportunity weapons rather than mere obstacles. So, to count as obstacles, they would have to be neutral non-capturing pieces that are either capturable by either player or movable by either player.
Whether capturable or mobile, obstacles should be thought of as neutral pieces that do not capture. If they are neither capturable nor mobile, then they are just features of the terrain like holes in the board and shouldn't properly be thought of as obstacles. If they can be moved by only one player and not the other, they would count as part of that player's forces rather than as obstacles. Likewise, if they could be captured by only one player, they could be considered part of the other player's forces in some sense. If they could capture other pieces, they would be equal-opportunity weapons rather than mere obstacles. So, to count as obstacles, they would have to be neutral non-capturing pieces that are either capturable by either player or movable by either player.