The Twins bullpen has five members returning:
Glen Perkins, closer,
Jarad Burton, primary setup,
Brian Duensing, lefty setup
Alex Burnett, middle relief
Anthony Swarzak, long relief
Swarzak just broke several ribs and may or may not be ready by opening day. Had he stayed healthy, he might have had a shot at the fifth spot in the rotation. But assuming he gets healthy, the Twins will need to find at least two more guys to fill out the bullpen.
They have a lot of options:
Casey Fien and Tyler Robertson are pitchers from last year that are still on the roster. That is a sign the Twins still see a future for them in the organization. Robertson is probably the leading candidate for a spot lefty, aka LOOGY. If Fien pitches like last year, he will nail down a spot.
Ryan Pressly was taken in the rule 5 draft from the Red Sox and must stay with the team or be returned to the Red Sox. He is supposed to be a hard thrower and will likely be given every chance to win a spot.
The Twins added a couple of candidates with major league experience to the major league roster. Tim Wood was signed to a major league deal by the Twins to avoid making him eligible to other teams in the Rule 5 draft. That is a sign of the talent the Twins hope to draw out of him. In addition to Wood, the Twins picked up Josh Roenicke on waivers.
The Twins have a few young guys on the roster that are probably a ways away. Caleb Thielbar is probably the one who is most likely to be competitive in spring training.
There are also a number of non-roster invitees. The most interesting is probably Rich Harden. Harden has had a lot of success in the major leagues when healthy. Unfortunately he is never healthy for long. He will likely start out by competing for a rotation spot, but may end up in the bullpen if the Twins think he is more likely to stay healthy there.
One thing that is important to remember. The failure of the Twins rotation last year put a lot of weight on the bullpen. If their starters go deeper into games, they could get as many as 200 innings more innings this year. Most of those innings will likely replace outs that came from the teams worst relievers last year. That would mean better overall numbers from the bullpen.