The Senate is often a conundrum wrapped in complexity, challenging the wherewithal of successive governments for many decades.

Unlike the House of Representatives, it is rare for the government of the day to control the red room. In fact, the last time it happened was during John Howard's final term as prime minister — and we know that didn't end well for the Liberal leader.

The Senate crossbench swelled to 11 members during the last term of government, after a double dissolution election was launched in an unsuccessful attempt to clear their ranks.

Malcolm Turnbull and his successor Scott Morrison had to court nine of them each and every time the Coalition wanted to press legislation without the support of Labor and the Greens.

The task facing Mr Morrison in the new Parliament is slightly easier, but still not a cakewalk. The crossbench comprises six members, and the Coalition will need to woo four of them to pass its agenda.

So, who are they?

There are some familiar faces on the Senate crossbench.

Pauline Hanson and Malcolm Roberts are in the Senate.

Two hail from Queensland — Pauline Hanson is still there, as leader of One Nation. Her lieutenant Malcolm Roberts returns to the red leather, following a High Court-imposed exile after he was involved in the dual citizenship fiasco.

Another former foreigner, Jacqui Lambie, also returns having secured one of Tasmania's Senate spots.

The South Australian politicians formerly known as the Nick Xenophon Team, the Centre Alliance, have retained their two spots — Rex Patrick and Stirling Griff occupying what they often describe as the "sensible centre".

Senators Patrick, Griff and Lambie have a "loose" arrangement when it comes to voting as a bloc on certain issues.

Rounding out the ranks is the leader of the Australian Conservatives, Adelaide-based Cory Bernardi. He wasn't up for re-election this time around, and remains in the Upper House.

Without the likes of former senators David Leyonhjelm and Fraser Anning, Senator Bernardi no longer has a bloc to back. It's understood he's considering his future, potentially flirting with the possibility of rejoining his former Liberal colleagues.

With no Fraser Anning (front), Cory Bernardi could rejoin the Liberals. ( ABC News: Marco Catalano )

Taxing times

The Coalition wants all three stages of its income tax cuts passed in full.

The Greens say "no". Labor is considering its options — arguments about the risks of locking in long-term tax cuts through to 2024/25 in the face of economic uncertainty are jostling against calls to pass the bill and seek to wind back the cuts later if circumstances change.

This is where the crossbench comes in. Senator Bernardi would back the Coalition, Senator Hanson says she will not.

The Centre Alliance senators from South Australia look likely to side with the Government if they can get a plan to combat rising gas prices.

So all the attention could turn to Senator Lambie. After time in the political wilderness, she's so far keeping her cards close to her chest.

The three South Australians and the Tasmanian would give the Coalition the numbers needed to get the cuts through the Senate.

The often unpredictable Jacqui Lambie could be key to passing some legislation. ( ABC News: Jessica Hayes )

Medevac malaise

The so-called "medevac" legislation was foisted upon the Coalition in February after Labor, the Greens and the crossbench banded together to change the process for medical transfers from Australia's offshore detention facilities.

The historic loss, the first for a government in decades, happened while Mr Morrison governed in minority. Fresh from his election win, he wants to repeal the legislation.

It will sail through the Lower House. Again, the Senate crossbench will come into play.

Centre Alliance was in favour of the legislation, and is unlikely to shift. One Nation would want to repeal, as would Senator Bernardi.

Again, all eyes will be on the often unpredictable Senator Lambie.