You know, there's a beautiful Spanish song which the Chilean folklorist Rolando Alarcón did a cover. He, as a communist militant and supporter of the Unidad Popular and Salvador Allende, made a whole disc with covers of songs from the Spanish Civil War era. I comment this as a little introduction, because if the Spanish Civil War was important for the Chilean left around 1969-1973 (and I'd argue from way before), certain characteristics of the Francoist ideology and regime were deeply debated for decades by the national-conservative intellectuality and -lastly- introduced to the Chilean cultural and legal panorama by the Civic-Military Regime of Pinochet.

I base my answer on the works of Isabel Jara Hinojosa, De Franco a Pinochet: El proyecto cultural franquista en Chile 1936-1980, (2006), and Carlos Huneeus, El régimen de Pinochet (2016). But I'll take mainly the first once since it goes deeply into the matter of the francoist influence on the regime.

As I previously stated, the national-conservative Chilean intellectuality of the first half of the XX Century was really interested on the values promoted by the Francoist foreign policy through its cultural approach: The Instituto de Cultura Hispánica (ICH) began to work in Latin America (and specifically in Chile) since 1946, opening its door to promote the sense of Hispanidad, as understood like the group of philosophical and cultural values (like its religious dogmatism, the heroic mythology, aristocratic nostalgia, nationalist exaltation) and fascist thoughts (like anticommunism, antiliberalism and corporatism) that the Francoist regime mixed.

Now, Isabel Jara goes into more details on what happened before the dictatorship, but to summarize what happened before: Even though renown names in the national-conservative Chilean academia (like Jaime Eyzaguirre and Osvaldo Lira) joined the ICH and even created a Chilean private institution to help the efforts of the ICH's political agenda, the reality clashed since Chile had a plural political landscape, and it wasn't well seen to support Franco by some parts of the society. Nonetheless, they began to adapt the Francoist Hispanidad to the Chilean landscape from their intellectual jobs (as historians, philosophers and universitary teachers). As stated by both authors, it's in this period in which the -then- student of Law Jaime Guzmán (main ideologist of the Pinochet dictatorship) studied and took these ideals from both Eyzaguirre and Lira (Guzmán was their student in the Universidad Católica).

Returning to 1973 and the beginning of the dictatorship, Carlos Huneeus comments that Pinochet saw on Franco an example to follow: Mainly the anticommunism of spanish dictatorship and the longevity of its reign. His regime suppressed think tanks, study centres and most of the cultural and intellectual panorama that existed in Chile at the time, but his support to Franco created a revival of the ICH and it's Chilean circle which began to have way more activity than before, as states Jara, at the point that they reached its influence into the educational program of the Chilean Ministry of Education (they already did it partially and to partially public school through donations, but their influence grew up from now on).

Not only that, days after the military coup, Jaime Guzmán (who was a really big opposition leader against Salvador Allende) became part of the Comisión Ortúzar, the commission that the dictatorship created to write a new Constitution for Chile. There, Guzmán took the thesis of Osvaldo Lira (influenced by the francoist ideals) in which it was proposed that what legitimize the nation and its rulers are the tradition and not the popular will. In this sense, Guzmán and the other intellectuals of the Commission, inspired on the Francoist regime but also aware of the dismantlement of the Spanish dictatorship after the decease of Francisco Franco, began to create ways in which the new pillars of Chile could continue for decades after the Armed Forces would leave the power and the political parties would return to the legality.

Returning to the ideals, the Military Junta identified in the Francoist influences ways to justify the existence and longevity of the regime as Jara states. In their Declaration of Principles of 1974, they identified themselves as the defenders of the national Catholic roots against the Marxism, hence they were the saviours of the country. As both authors point out, the dictatorship identified itself as Economically (Neo) Liberal and Conservative in its values. Interestingly enough, this definition began a heated debate inside the collaborators of the regime: The traditionalist influenced by the fascist ideals were unhappy that the corporatism ideals weren't taken into account outside of the moralist part and that Pinochet followed the path and advice created by the Chicago Boys neoliberal economists.

Even with that fracture, Jara says that they still supported the regime and worked in the National Secretary of the Women and the National Secretary of the Youth. In both offices it was promoted that the traditional and authoritarian catholic family was the core of the new society, and rooted Christian and National values to the youth, this in order to create a new professional elite that was apolitical.

Heading to the central figure of Pinochet, Huneeus comments that he felt identified with the Francoist regime through its Anticommunist roots and the longevity of Franco's regime (specifically the image of perpetuity of the regime). Talking about the funeral of Franco, Pinochet was one of the few Heads of State that attended it, but he couldn't participate in the assumption of Juan Carlos due to the rejection of most European Heads of State who threatened to not attend the ceremony if he participated.

Most importantly, Huneeus makes a comparison between both regimes, stating that in both cases there's a central dictatorial figure associated to the regime (Franco stayed in power for 36 years and Pinochet for 17 years). The economical modernization of both countries happened under authoritarianism, being examples of Prussian modernization, a modernization that happens without public liberties nor the presence of the Rule of Law. Both countries also opened their economies to the market during the regimes. But they begin to differ precisely in the economical term, that's because the Spanish technocrats who opened the market in Spain were a heterogeneous group of different political affiliations (specially after the decease of Franco) that proposed different kinds of policies to surpass the economical stagnation, weren't united (even fought on their different projects), while the Chilean Chicago Boys identified themselves with the extreme neoliberalism, adhered politically and fully to the dictatorship and had no opposition to their economical plans, even less the existence of economical alternatives.

Lastly, I would like to point out that there were other influences that the spanish dictatorship (and the other fascist regimes) gave to the Chilean one, but from what I have already studied and re-read for this response, these were the ones I could answer for. I haven't seen any related to the sharing of tactics in terms of suppresion. or other militaristic ways in which Franco could have collaborated directly with Pinochet. Nothing closer to what the Operation Cóndor did. But, there were some exiled nazis working for the Chilean regime, though, like Walter Rauff and Paul Schäfer.

I hope this can help with your answer! I will edit some grammar mistakes soon. You can check some other answers I have given on the Pinochet's regime here.

Sources:

• Huneuus, Carlos (2016). El Régimen de Pinochet.
• Jara, Isabel (2006). De Franco a Pinochet: El proyecto cultural franquista en Chile 1936-1980.